#  Loop quantum deparametrized Schwarzschild interior and discrete black hole
mass

Cong Zhang111<EMAIL_ADDRESS>Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland  Yongge Ma222<EMAIL_ADDRESS>Department of
Physics, Beijing Normal University, Beijing 100875, China  Shupeng
<EMAIL_ADDRESS>School of Physics, Beijing Institute of Technology,
Beijing 100081, China  Xiangdong Zhang444<EMAIL_ADDRESS>Department of
Physics, South China University of Technology, Guangzhou 510641, China

###### Abstract

We present the detailed analyses of a model of loop quantum Schwarzschild
interior coupled to a massless scalar field and extend the results in our
previous rapid communication Zhang et al. (2020) to more general schemes. It
is shown that the spectrum of the black hole mass is discrete and does not
contain zero. This supports the existence of a black hole remnant after
Hawking evaporation due to loop quantum gravity effects. Besides to show the
existence of a non-vanishing minimal black hole mass in the vacuum case, the
quantum dynamics for the non-vacuum case is also solved and compared with the
effective one.

###### pacs:

04.60.Pp, 04.70.Dy

##  I Introduction

As a prediction of general relativity (GR), the existence of black holes has a
broad base of support from observations Akiyama et al. (2019). However, our
understanding on black hole (BH) is still far from the end. Among those
challenging topics on BH, its quantum nature is particularly interesting. By
studying quantum BHs, one could not only solve puzzles originating from the
classical theory, but also achieve more understandings on the theory of
quantum gravity.

As a background independent approach to quantum gravity, loop quantum gravity
(LQG) has been widely studied in the past 30 years Ashtekar and Tate (1991);
Rovelli (2004); Ashtekar and Lewandowski (2004); Han et al. (2007); Thiemann
(2008); Abhay and Jorge (2017). Although some important breakthroughs have
been made in LQG Rovelli and Smolin (1995); Ashtekar et al. (1995); Ashtekar
and Lewandowski (1997a, b); Thiemann (1998a, b); Ma and Ling (2000);
Lewandowski et al. (2006); Yang and Ma (2016), its dynamics is still an open
issue. The obstacle of LQG can be bypassed through applying the loop
quantization techniques to the symmetry-reduced sectors of GR, where the
expression of the Hamiltonian constraint becomes much simpler than that in the
full theory. The resulting quantum models are expected to reflect some quantum
features of full LQG, in spite of the fact that they might not be equivalent
to the direct symmetric sector of full LQG. An improved treatment of quantum-
reduced LQG has also been proposed to study the symmetric sectors Alesci et
al. (2013). These ideas were applied to study loop quantum Schwarzschild BH
recently with different perspectives Ashtekar and Bojowald (2005); Modesto
(2006); Boehmer and Vandersloot (2007); Campiglia et al. (2008); Chiou (2008);
Gambini and Pullin (2013); Corichi and Singh (2016); Bojowald et al. (2018);
Ashtekar et al. (2018a, b); Rovelli and Vidotto (2018); Bianchi et al. (2018);
Achour et al. (2018); Alesci et al. (2018, 2019); Bodendorfer et al. (2019a,
b); Kelly et al. (2020); Han and Liu (2020). However, most of these studies
focused on the effective dynamics, where one considered the Hamiltonian
constraint with the holonomy correction and solved the effective Hamiton’s
equations. This treatment resulted in several important achievements. In
particular, it resolves the singularity inside the Schwarzschild BH and
predicts certain extensions of the Schwarzschild interior beyond the
singularity (see, for instance, Ashtekar et al. (2018a, b); Bodendorfer et al.
(2019b); Kelly et al. (2020); Han and Liu (2020)). However, in the effective
prescription one cannot see more intrinsic quantum natures of BH, such as the
ground state of quantum BHs and the discreteness of the spectrum of Dirac
observables. After all, it is necessary to consider the quantum dynamics.

There are several crucial topics on the quantum dynamics of BH. One is the
issue of the final state of BH evaporation which is related to the constituent
of dark matter and the puzzle of information loss. According to the Hawking
radiation Hawking (1974), the primordial mini BHs in the very early universe
should be completely evaporated by now. However, if the BH evaporation is
halted at some stable state by some quantum gravity effect, which is called
the BH remnant, these remnants would result in important cosmological
consequences Barrow et al. (1992); Carr et al. (1994); Dalianis and Tringas
(2019). Remarkably, the remnants originating from these primordial BHs could
even comprise the entire dark matter in the universe Barrow et al. (1992);
Carr et al. (1994). Moreover, thanks to the remnant, one could argue that the
information fallen into a BH with matters could be stored in the remnant after
its evaporation. This provides a possible approach to solve the puzzle of
information loss Preskill (1992); Chen et al. (2015). Furthermore, the
distortion of the semiclassical Hawking spectrum resulted from certain
discreteness of the BH mass was studied Bekenstein (1997); Pranzetti (2012);
Barrau et al. (2015); Lochan and Chakraborty (2016). It was argued that in
certain cases, the distortion could be observable even for macroscopic BHs
Bekenstein (1997). Although these debates are crucial and long standing, there
was no systematic study by quantum gravity to lay a solid theoretical
foundation for the arguments until the prediction of a BH remnant in LQG
models by Zhang et al. (2020). The purpose of this paper is to provide the
detailed constructions in Zhang et al. (2020) and extend the results to more
general schemes. Moreover, the quantum dynamics of the model will be further
studied in detail.

We study the model of loop quantum Schwarzschild interior coupled to a
massless scalar field. The quantum Schwarzschild BH, as the vacuum case of
this model, can be resulted by vanishing the scalar field. The phase space of
this system contains three pairs of canonical variables, $(b,p_{b})$ and
$(c,p_{c})$ for gravity, and $(\varphi,p_{\varphi})$ for the scalar field. By
deparametrizing this model, one gets the physical Hamiltonian
$\sqrt{\mathfrak{h}}$ of the relational evolution with respect to the scalar
field $\varphi$ Alesci et al. (2015); Dapor et al. (2013); Zhang and Zhang
(2020). In the classical theory, the Poisson bracket between $\mathfrak{h}$
and the ADM mass $M$ vanishes. This indicates a classical Dirac observable
$m=cp_{c}$ proportional to $M$. However, this commutativity may no longer be
kept by the corresponding operators $\hat{\mathfrak{h}}$ and $\hat{m}$ in the
quantum theory, which is relevant to the choice of schemes for the
quantization of $\mathfrak{h}$. We only focus on the schemes such that
$\hat{\mathfrak{h}}$ and $\hat{m}$ are still commutative, since the
commutativity means the existence of a Dirac observable $\hat{m}$. Note that a
general class of schemes adopted for the loop quantization of the current
model can meet our requirement. In particular, it is valid for the
$\mu_{o}$-scheme Ashtekar and Bojowald (2005); Modesto (2006) and the new
scheme balancing the $\mu_{o}$ scheme and $\bar{\mu}$ scheme Corichi and Singh
(2016); Ashtekar et al. (2018a, b). However, it cannot be met by the
$\bar{\mu}$ scheme Boehmer and Vandersloot (2007); Chiou (2008).

We will first construct the Hamiltonian operator $\hat{\mathfrak{h}}$ and
study its properties analytically. Thanks to these analytical results, a
numerical method to diagonalize $\hat{\mathfrak{h}}$ is proposed so that the
dynamics is computable. Then the quantum dynamics of the model can be solved
for the non-vacuum case, and it can be compared with the effective one. For
the vacuum case, the Hilbert space consisting of the physical states of the
Schwarzschild BH is built up. In this Hilbert space $m$ is promoted to an
operator which has discrete spectrum $\overline{\sigma_{\xi}}$ with
$0\notin\overline{\sigma_{\xi}}$. This result supports the existence of a
stable BH remnant.

This paper is organized as follows. In Sec II, the theory of loop quantum
Schwarzschild BH interior coupled to a massless scalar field is briefly
reviewed, including the deparametrization and the polymer quantization of this
model. In Sec. III, we construct the physical Hamiltonian operator and study
its properties analytically. Then the quantum dynamics for both non-vacuum and
vacuum cases are solved in Sec. V. Finally, in Sec. VI, our results are
summarized and discussed.

##  II Preliminaries

###  II.1 Deparametrization of the model

Given a spatially homogeneous 3-manifold $\Sigma$ of topology
$\mathbb{R}\times S^{2}$. Because of the homogeneity, $\Sigma$ is endowed with
a fiducial metric

$\mathring{q}_{ab}{\rm d}x^{a}{\rm d}x^{b}={\rm d}x^{2}+r_{o}^{2}({\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2})$ (1)

where $(x,\theta,\phi)$ are the natural coordinates adapted to the topology,
and $r_{o}$ is a constant with dimension of length. Since $\Sigma$ is non-
compact in the $x$ direction, we introduce an elementary cell
$\mathcal{C}\cong(0,L_{0})\times\mathbb{S}^{2}$ in $\Sigma$ and restrict all
integrals to this elemental cell to avoid the divergence of integrations.

The classical phase space of gravity coupled to a massless scalar field
contains the Ashtekar-Barbero canonical conjugate pairs
$(A_{a}^{i}(x),E^{a}_{i}(x))$ for gravity and $(\varphi(x),\pi(x))$ for scalar
field. As far as the homogeneous states are concerned, the scalar filed
$\varphi$ is reduced to a constant and the fields $A_{a}^{i}(x)$,
$E^{a}_{i}(x)$ and $\pi_{\varphi}(x)$ take the forms Zhang and Zhang (2020)

$\displaystyle A_{a}^{i}\tau_{i}{\rm d}x^{a}$
$\displaystyle=\frac{c}{L_{0}}\tau_{3}{\rm d}x+b\tau_{2}{\rm
d}\theta-b\tau_{1}\sin\theta{\rm d}\phi+\tau_{3}\cos\theta{\rm d}\phi$ (2)
$\displaystyle E_{i}^{a}\tau^{i}\partial_{a}$
$\displaystyle=p_{c}\tau_{3}\sin\theta\partial_{x}+\frac{p_{b}}{L_{0}}\tau_{2}\sin\theta\partial_{\theta}-\frac{p_{b}}{L_{0}}\tau_{1}\partial_{\phi}$
$\displaystyle\pi_{\varphi}$ $\displaystyle=\frac{p_{\varphi}}{4\pi
r_{0}^{2}}\sqrt{\mathring{q}}=\frac{p_{\varphi}}{4\pi}\sin\theta$

where $\tau_{j}=-i\sigma_{j}/2$ $(j=1,2,3)$ with $\sigma_{j}$ being the Pauli
matrix and, $c,\ b,\ p_{c},\ p_{b}$ and $p_{\varphi}$ are all constants.
According to (2), the symmetry-reduced phase space is coordinatized by the
pairs $(c,p_{c})$, $(b,p_{b})$ and $(\varphi,p_{\varphi})$. The non-vanishing
Poisson brackets read

$\\{c,p_{c}\\}=2G\gamma,\ \\{b,p_{b}\\}=G\gamma,\
\\{\varphi,p_{\varphi}\\}=1.$ (3)

By the symmetry-reduced expression (2), the Gaussian and diffeomorphism
constraints vanish automatically. The dynamics of this model is encoded in the
Hamiltonian constraint. In the full theory of gravity coupled to a massless
scalar field, the Hamiltonian constraint can be deparametrized as Domagała et
al. (2010)

$\displaystyle C(x)=\pi_{\varphi}(x)\pm\sqrt{h(x)}=0,$ (4)

where $h(x)=-2\sqrt{|\det(E(x))|}C_{\rm gr}(x)$, with $C_{\rm gr}$ being the
vacuum-gravity Hamiltonian constraint. The so-called physical Hamiltonian can
be written as

$h_{\rm phy}=\frac{\sqrt{G}\gamma}{\sqrt{4\pi}}\int_{\Sigma}{\rm
d}^{3}x\sqrt{h(x)}=\frac{\gamma}{4\sqrt{2}\pi}\int_{\Sigma}{\rm
d}^{3}x\sqrt{E_{i}^{a}E_{j}^{b}\left(-F^{k}_{ab}\epsilon_{ijk}+2(1+\gamma^{2})K_{[a}^{i}K_{b]}^{j}\right)}$
(5)

where $F^{k}_{ab}$ denotes the curvature of the connection $A_{a}^{i}$,
$K_{a}^{i}$ is the extrinsic curvature and the prefactor is adapted for
convenience. It generates the relational evolution with respect to the scalar
field. Substituting (2) in to (4) and integrating both sides in the elementary
cell $\mathcal{C}$, one finally obtains the Hamiltonian constraint of our
model as

$p_{\varphi}\pm\frac{\sqrt{4\pi}}{\sqrt{G}L_{0}\gamma}\sqrt{\mathfrak{h}}=0$
(6)

where $\mathfrak{h}$ is given by

$\mathfrak{h}:=p_{b}\left(\left(b^{2}+\gamma^{2}\right)p_{b}+2bcp_{c}\right)$
(7)

In this work, we will also refer to the physical Hamiltonian as $\mathfrak{h}$
though it is related to the true physical Hamiltonian $h_{\rm phy}$ by
$\sqrt{\mathfrak{h}}=h_{\rm phy}$.

The quantum states of the model are described by vectors in the Hilbert space
$\mathcal{H}_{t}=\mathcal{H}_{\rm mat}\otimes\mathcal{H}_{\rm gr}$, where
$\mathcal{H}_{\rm mat}$ and $\mathcal{H}_{\rm gr}$ are the Hilbert spaces
describing the matter and gravity respectively. The physical states
$|\psi\rangle_{\rm phy}$ should satisfy the quantum version of the Hamiltonian
constraint, i.e.,

$\left(\hat{p}_{\varphi}\pm\frac{\sqrt{4\pi}}{L_{0}\gamma\sqrt{G}}\widehat{\sqrt{\mathfrak{h}}}\right)|\psi\rangle_{\rm
phy}=0.$ (8)

It gives us a Schödinger-like equation. Therefore, in the case that the
operator $\widehat{\sqrt{\mathfrak{h}}}$ is self-adjoint, the physical states,
i.e., the solutions of (8), can be expressed as

$|\psi\rangle_{\rm phy}=e^{\mp i\frac{\sqrt{4\pi
G}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi\widehat{\sqrt{\mathfrak{h}}}}|\psi\rangle_{\rm
gr}$ (9)

where $|\psi\rangle_{\rm gr}\in\mathcal{H}_{\rm gr}$ is a state of pure
gravity and $\ell_{\mathrm{p}}:=\sqrt{G\hbar}$ is the Plank length. A Dirac
observable $\mathcal{O}_{\rm phy}$ takes the form

$\mathcal{O}_{\rm phy}=e^{\mp i\frac{\sqrt{4\pi
G}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi\widehat{\sqrt{\mathfrak{h}}}}\mathcal{O}_{\rm
gr}e^{\pm i\frac{\sqrt{4\pi
G}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi\widehat{\sqrt{\mathfrak{h}}}}$
(10)

with an operator $\mathcal{O}_{\rm gr}$ in $\mathcal{H}_{\rm gr}$.

To carry out the above deparametrization procedure, in next subsection we will
introduce the polymer quantization for the gravity and obtain its Hilbert
space $\mathcal{H}$. Then a self-adjoint operator $\hat{\mathfrak{h}}$ on
$\mathcal{H}$ is proposed by the loop quantization procedure. Finally we
restrict ourselves to the subspace consisting of non-negative spectra of
$\hat{\mathfrak{h}}$ to define the Hilbert space

$\mathcal{H}_{\rm gr}:=\hat{P}_{[0,\infty)}\mathcal{H}$ (11)

where $\hat{P}_{[0,\infty)}:=\chi_{[0,\infty)}(\hat{\mathfrak{h}})$ is the
projection operator with respect to the spectrum decomposition of
$\hat{\mathfrak{h}}$.

###  II.2 The polymer quantization

The polymer quantization of the gravity in this model leads to the Hilbert
space Ashtekar and Bojowald (2005)

$\tilde{\mathcal{H}}=\tilde{\mathcal{H}}_{\mathfrak{b}}\otimes\tilde{\mathcal{H}}_{\mathfrak{c}}=L^{2}(\mathbb{R}_{\rm
Bohr},{\rm d}\mu_{0})\otimes L^{2}(\mathbb{R}_{\rm Bohr},{\rm d}\mu_{0}),$
(12)

where ${\rm d}\mu_{0}$ is the Haar measure on the Bohr compactification
$\mathbb{R}_{\rm Bohr}$ of the real line $\mathbb{R}$ (Thiemann, 2008, Chapter
28). The two spaces $\tilde{\mathcal{H}}_{\mathfrak{b}}$ and
$\tilde{\mathcal{H}}_{\mathfrak{c}}$ correspond to the canonical conjugate
pairs $(b,p_{b})$ and $(c,p_{c})$ respectively. The standard basis of these
two Hilbert spaces are denoted by
$|\mu\rangle\in\tilde{\mathcal{H}}_{\mathfrak{b}}$ and
$|\tau\rangle\in\tilde{\mathcal{H}}_{\mathfrak{c}}$. Their inner products read

$\langle\mu^{\prime}|\mu\rangle=\delta_{\mu^{\prime},\mu},\
\langle\tau^{\prime}|\tau\rangle=\delta_{\tau^{\prime},\tau},$ (13)

where the Kronecker-$\delta$ symbol is employed.

There are two types of basic operators in the Hilbert spaces. One is the
momentum-variable operators $\hat{p}_{b}$ and $\hat{p}_{c}$, whose actions on
$|\mu\rangle$ and $|\tau\rangle$ are given by

$\displaystyle\hat{p}_{b}\,|\mu\rangle$
$\displaystyle=\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}\mu|\mu\rangle$ (14)
$\displaystyle\hat{p}_{c}\,|\tau\rangle$
$\displaystyle=\gamma\ell_{\mathrm{p}}^{2}\,\tau|\tau\rangle.$

The other is the configuration-variable operators $\widehat{e^{i\lambda b}}$
and $\widehat{e^{i\lambda c}}$, whose actions read

$\displaystyle\widehat{e^{i\lambda b}}|\mu\rangle$
$\displaystyle=|\mu+2\lambda\rangle$ (15) $\displaystyle\widehat{e^{i\lambda
c}}|\tau\rangle$ $\displaystyle=|\tau+2\lambda\rangle.$

As in the model of loop quantum cosmology (LQC), the operators
$\widehat{e^{i\lambda b}}$ and $\widehat{e^{i\lambda c}}$ correspond to the
holonomies along the edges parallel to the $\mathbb{R}$ direction and the
equator (or the longitude because of the homogeneity) of $\mathbb{S}^{2}$
respectively. Moreover, $\widehat{e^{i\lambda b}}$ and $\widehat{e^{i\lambda
c}}$ are not strongly continuous with respect to $\lambda$. Therefore there
are no operators corresponding to the configuration variables $b$ and $c$ in
our model. This respect to the fact that there does not exist operator
corresponding to the connection itself in the full theory.

##  III The physical Hamiltonian operator

###  III.1 A separable subspace $\mathcal{H}$ of $\tilde{\mathcal{H}}$

According to (13) , the Hilbert space $\tilde{\mathcal{H}}$ possesses a non-
countable orthonormal basis, and hence it is non-separable. The problem that
the kinematical Hilbert space is non-separable has appeared in both LQG and
LQC. In LQG, the non-separability is caused by the uncountability of the
graphs based on which the Hilbert space is defined. In LQC, it is caused by
the polymer quantization procedure which leads to an uncountable orthonormal
basis as the case of the present model. In LQG, one proposed to employ the
diffeomorphism invariance to identify the diffeomorphism equivalent graphs in
order to obtain a separable diffeomorphism invaraint Hilbert space Ashtekar
and Lewandowski (2004). In LQC, the problem is tackled by the super-selection
feature of the Hamiltonian constraint. More precisely, because the Hamiltonian
constraint operator preserves some separable Hilbert subspaces, one can
confine the study in certain separable subspace Ashtekar et al. (2003, 2006).
The situation in the present model is very similar to that in LQC. As shown
below, a separable subspace $\mathcal{H}\subset\tilde{\mathcal{H}}$ can be
selected, which is preserved by the Hamiltonian operator.

Classically, the physical Hamiltonian (7), re-denoted by $\mathfrak{h}_{c}$,
can be rewritten as

$\mathfrak{h}_{c}=2p_{b}bp_{c}c+p_{b}^{2}b^{2}+\gamma^{2}p_{b}^{2},$ (16)

where $p_{c}c$ is a Dirac observable. Actually, for vacuum gravity, one has
$p_{c}c=L_{0}\gamma GM$ with $M$ being the ADM mass of Schwarzschild BH Zhang
and Zhang (2020). Since in the Hilbert space $\tilde{\mathcal{H}}$ there are
no operators corresponding to $b$ and $c$, the expression (16) cannot be
promoted to an operator directly. We thus return to the integral expression
(5) of the full deparametrized theory and follow Ashtekar and Bojowald (2005)
to express $C_{\rm gr}$ in terms of $\mathring{F}={\rm d}(\gamma K)+[\gamma
K,\gamma K]$ and the spatial curvature $\Omega=-\sin(\theta){\rm
d}\theta\wedge{\rm d}\phi\tau_{3}$, with

$K=\frac{1}{\gamma}\left(\frac{c}{L_{0}}\tau_{3}{\rm d}x+b\tau_{2}{\rm
d}\theta-b\tau_{1}\sin\theta{\rm d}\phi\right)$ (17)

denoting the extrinsic curvature. Then the physical Hamiltonian $h_{\rm phy}$
is expressed by

$h_{\rm phy}=\frac{1}{4\sqrt{2}\pi}\int_{\mathcal{C}}{\rm
d}^{3}x\sqrt{E^{a}_{i}(x)E^{b}_{j}(x)(\mathring{F}^{k}_{ab}(x)-\gamma^{2}\Omega_{ab}^{k}(x))}.$
(18)

Note that, in comparison with the expression (2) of $A$, the expression (17)
does not contain the term $\tau_{3}\cos(\theta){\rm d}\phi$. Thus the
“holonomy” of $K$ along the $\phi$-direction is much simpler than that of $A$.
Moreover, since $\Omega$ does not depend on dynamical variables, one needs not
to regularize it by holomolies. To regularize (18), one fixes three edges
intersecting at a point $x_{0}\in\mathcal{C}$, where the first edge $e_{1}$ is
along the $\mathbb{R}$ direction of $\Sigma$ taking length
$\ell_{1}=\delta_{c}L_{0}$ and the other two edges $e_{2}$ and $e_{3}$ are
along the equator and the longitude of $\mathbb{S}^{2}$ with the same radians
$\ell_{2}=\ell_{3}=\delta_{b}$. By defining the “holonomies”

$h_{i}=\exp\left(\int_{e_{i}}K\right)$ (19)

one can regularize $\mathring{F}_{ab}^{k}$ as Ashtekar and Bojowald (2005)

$\mathring{F}_{ab}^{k}(x_{0})=-\sum_{i,j}\frac{2}{\ell_{i}\ell_{j}}\mathrm{tr}\left(h_{i}h_{j}h_{i}^{-1}h_{j}^{-1}\tau_{k}\right)\mathring{\omega}_{a}^{i}\mathring{\omega}_{b}^{j}+O(\sqrt{\ell_{a}\ell_{b}})$
(20)

where $\mathring{\omega}_{a}^{i}$ denotes the co-triad of the fiducial metric
(1) , adapting to the three edges $e_{i}$. Substituting (20) and the
expression (2) of $E^{a}_{i}$ into (18), we thus get a regularized version of
$h_{\rm phy}=\sqrt{\mathfrak{h}}$ for our model, where

$\mathfrak{h}=2p_{b}\frac{\sin(\delta_{b}b)}{\delta_{b}}p_{c}\frac{\sin(\delta_{c}c)}{\delta_{c}}+p_{b}^{2}\frac{\sin^{2}(\delta_{b}b)}{\delta_{b}^{2}}+\gamma^{2}p_{b}^{2}.$
(21)

The expression (21) returns to $\mathfrak{h}_{c}$ in (16) at the limit of
$\delta_{b}\to 0$ and $\delta_{c}\to 0$.

To quantize the regularized physical Hamiltonian (21) instead of (16), various
strategies have been proposed in choosing the two quantum parameters
$\delta_{b}$ and $\delta_{c}$. Roughly speaking, the choices can be classified
into three schemes. The first one is the $\mu_{0}$-scheme where $\delta_{b}$
and $\delta_{c}$ are chosen as constants Ashtekar and Bojowald (2005); Modesto
(2006); Campiglia et al. (2008). The second one is the $\bar{\mu}$-scheme
which allows $\delta_{b}$ and $\delta_{c}$ to be any functions of $p_{b}$ and
$p_{c}$ Boehmer and Vandersloot (2007); Chiou (2008). The third one, referred
to as the modified scheme, is developed recently where $\delta_{b}$ and
$\delta_{c}$ are phase space dependent only through Dirac observables Corichi
and Singh (2016); Ashtekar et al. (2018a, b). Since the expression
$p_{c}\sin(\delta_{c}c)/\delta_{c}$ in (21) corresponds to the classical Dirac
observable $p_{c}c=L_{0}\gamma GM$ in (16), we are motivated to assume that
the quantum operator $\hat{\mathfrak{h}}$ is composed of a Dirac observable
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$, which is self-adjoint and
commutes with $\hat{\mathfrak{h}}$. This assumption rules out certain
strategies such as the $\bar{\mu}$-scheme where $\delta_{c}$ depends on
$\hat{p}_{b}$ and $\delta_{b}$ depends on $\hat{p}_{c}$, since the resulting
operator would no longer commute with $\hat{\mathfrak{h}}$. Furthermore, we
assume that $\delta_{b}$ is a constant or any function of the Dirac observable
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ for following analysis. Similar
assumptions were adopted in $\mu_{o}$-scheme and the modified scheme. They are
sufficient but not necessary to obtain a Dirac observable
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$. In summary, the current paper
focuses on the schemes such that

  * (i)

a separable Hilbert subspace
$\mathcal{H}_{\mathfrak{c}}\subset\tilde{\mathcal{H}}_{\mathfrak{c}}$ can be
chosen to define an operator $\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$
corresponding to $p_{c}\sin(\delta_{c}c)/\delta_{c}$ which is self-adjoint,
commutates with the physical Hamiltonian $\hat{\mathfrak{h}}$.

  * (ii)

the quantum parameter $\delta_{b}$ is a constant or any function of
$p_{c}\sin(\delta_{c}c)/\delta_{c}$.

Denote $\sigma_{\mathfrak{c}}$ as the spectrum of the Dirac observable
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$. Then
$\mathcal{H}_{\mathfrak{c}}$ is isometric to the Hilbert space
$L^{2}(\sigma_{\mathfrak{c}},{\rm d}\mu_{\mathfrak{c}}))$ with the spectral
measure ${\rm d}\mu_{\mathfrak{c}}$. The Hilbert space
$\tilde{\mathcal{H}}_{\mathfrak{b}}\otimes\mathcal{H}_{\mathfrak{c}}\subset\tilde{\mathcal{H}}$
can be represented by $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\tilde{\mathcal{H}}_{\mathfrak{b}})$ in which each
element is a $\tilde{\mathcal{H}}_{\mathfrak{b}}$-valued function on
$\sigma_{\mathfrak{c}}$. The representation is defined by

$U:\tilde{\mathcal{H}}_{\mathfrak{b}}\otimes L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}})\ni\psi_{b}\otimes\psi_{c}\mapsto\psi_{c}(\cdot)\psi_{b}\in
L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\tilde{\mathcal{H}}_{\mathfrak{b}}).$ (22)

The inner product in $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu;\tilde{\mathcal{H}}_{\mathfrak{b}})$ is given by

$\left(\psi^{(1)},\psi^{(2)}\right)=\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\langle\psi^{(1)}(x)|\psi^{(2)}(x)\rangle$ (23)

where $\langle\psi^{(1)}(x)|\psi^{(2)}(x)\rangle$ denotes the inner produce of
$\psi^{(1)}(x),\psi^{(2)}(x)\in\tilde{\mathcal{H}}_{\mathfrak{b}}$.

For convenience, the elements in the spectrum space can be denoted by
$L_{o}\gamma m\in\sigma_{\mathfrak{c}}$ with $m\in\mathbb{R}$ in analogy with
their classical correspondence $p_{c}c=L_{o}\gamma GM$. We also define

$\hat{\beta}_{\lambda}:=\widehat{p_{b}\sin(\lambda b)}.$ (24)

For a state $\psi\in L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\tilde{\mathcal{H}}_{\mathfrak{b}})$, $\psi(L_{o}\gamma
m)\in\tilde{\mathcal{H}}_{\mathfrak{b}}$ is abbreviated to $\psi(m)$. Then the
action of $\hat{\mathfrak{h}}$ on $\psi$ reads

$(\hat{\mathfrak{h}}\psi)(m)=\left(\frac{2L_{o}\gamma
m}{\delta_{b}^{(m)}}\hat{\beta}_{\delta_{b}^{(m)}}+\frac{1}{(\delta_{b}^{(m)})^{2}}\hat{\beta}_{\delta_{b}^{(m)}}^{2}+\gamma^{2}\hat{p}_{b}^{2}\right)\psi(m)$
(25)

where $\delta_{b}^{(m)}\equiv\delta_{b}(L_{o}\gamma m)$ due to the dependence
of $\delta_{b}$ on $\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$. By (25), the
separable Hilbert subspace $\mathcal{H}\subset
L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\tilde{\mathcal{H}}_{\mathfrak{b}})$ preserved by
$\hat{\mathfrak{h}}$ is constructed as follows. Given $L_{o}\gamma
m\in\sigma_{\mathfrak{c}}$, let
$\mathcal{H}_{\mathfrak{b}}(m)\subset\tilde{\mathcal{H}}_{\mathfrak{b}}$ be a
separable Hilbert space preserved by $\hat{\beta}_{\delta_{b}^{(m)}}$. Then
the Hilbert space $\mathcal{H}\subset L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\tilde{\mathcal{H}}_{\mathfrak{b}})$ contains the state
$\psi$ such that $\psi(m)\in\mathcal{H}_{\mathfrak{b}}(m)$ for all
$L_{o}\gamma m\in\sigma_{\mathfrak{c}}$. For convenience, the Hilbert space
$\mathcal{H}$ constructed by
$\mathcal{H}_{\mathfrak{b}}(\cdot):m\mapsto\mathcal{H}_{\mathfrak{b}}(m)$
through this procedure will be denoted by $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot))$.

###  III.2 The operators $\hat{\beta}_{\lambda}$

According to (25), the action of $\hat{\mathfrak{h}}$ is determined by the
property of $\hat{\beta}_{\delta_{b}^{(m)}}$. Since $\delta_{b}^{(m)}$ is
constant for a given $m\in\mathbb{R}$, we will drop the superscript in
$\delta_{b}^{(m)}$ and consider $\hat{\beta}_{\delta_{b}}$ with a constant
$\delta_{b}$. Classically, one has

$\displaystyle p_{b}\sin(\delta_{b}b)$
$\displaystyle=\frac{1}{2i}\left(p_{b}e^{i\delta_{b}b}-p_{b}e^{-i\delta_{b}b}\right).$
(26)

A well-known ambiguity in the quantization procedure is caused by the operator
ordering. By definition, one has

$\widehat{e^{i\delta_{b}b}}\hat{p}_{b}=(\hat{p}_{b}-\delta_{b}\gamma\ell_{\mathrm{p}}^{2})\widehat{e^{i\lambda
b}}.$ (27)

We hereby introduce a parameter $\mathfrak{b}$ to parameterize various
operator-ordering strategies and define

$\widehat{p_{b}e^{i\delta_{b}b}}=(\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})\widehat{e^{i\delta_{b}b}}.$
(28)

Then, we can define a symmetric operator corresponding to (26) as

$\displaystyle\hat{\beta}_{\delta_{b}}$
$\displaystyle:=\frac{1}{2i}\left((\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})\widehat{e^{i\delta_{b}b}}-\widehat{e^{-i\delta_{b}b}}(\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})\right).$
(29)

Its action is given by

$\displaystyle\hat{\beta}_{\delta_{b}}|\mu\rangle$
$\displaystyle=\frac{1}{2i}\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}\Big{(}(\mu+2\delta_{b}+2\mathfrak{b})|\mu+2\delta_{b}\rangle-(\mu+2\mathfrak{b})|\mu-2\delta_{b}\rangle\Big{)}.$
(30)

Thus the separable Hilbert subspaces of $\tilde{\mathcal{H}}_{\mathfrak{b}}$
preserved by $\hat{\beta}_{\delta_{b}}$ are given by

$\displaystyle\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
$\displaystyle=\overline{\\{\psi\in\tilde{\mathcal{H}}_{\mathfrak{b}},\psi(\mu)\neq
0\text{ only for }\mu=\varepsilon_{b}+2n\delta_{b},n\in\mathbb{Z}\\}}$ (31)

for some constant $\varepsilon_{b}\in[0,2\delta_{b})$. We will show in below
that the separable Hilbert subspace preserved by the physical Hamiltonian
$\hat{\mathfrak{h}}$ can be constructed with
$\mathcal{H}_{\mathfrak{b}}^{\varepsilon_{b}}$.

Denote the restriction of $\hat{\beta}_{\delta_{b}}$ on
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ by
$\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$.
Given $\mathcal{H}_{\mathfrak{b}}^{\varepsilon_{b}}$ and
$\mathcal{H}_{\mathfrak{b}}^{\tilde{\varepsilon}_{b}}$, a natural isomorphism
between them can be defined by

$i:\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}\ni|\varepsilon_{b}+2n\delta_{b}\rangle\mapsto|\tilde{\varepsilon}_{b}+2n\delta_{b}\rangle\in\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}.$
(32)

It introduces an operator
$i(\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}$
in $\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ whose action reads

$i(\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}|\tilde{\varepsilon}_{b}+2n\delta_{b}\rangle=\frac{1}{2i}\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}\Big{(}(2n\delta_{b}+\tilde{\varepsilon}_{b}+2\delta_{b}+2\tilde{\mathfrak{b}})|\tilde{\varepsilon}_{b}+2n\delta_{b}+2\delta_{b}\rangle-(2n\delta_{b}+\tilde{\varepsilon}_{b}+2\tilde{\mathfrak{b}})|\tilde{\varepsilon}_{b}+2n\delta_{b}-2\delta_{b}\rangle\Big{)},$
(33)

where
$\tilde{\mathfrak{b}}:=\mathfrak{b}+(\varepsilon_{b}-\tilde{\varepsilon}_{b})/2$.
For clarity, let us use $\hat{\beta}_{\delta_{b}}^{(x)}$ to denote the
operator $\hat{\beta}_{\delta_{b}}$ with respect to the constant
$\mathfrak{b}=x$. According to (33), to study the operator
$\hat{\beta}_{\delta_{b}}^{(x)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
with respect to $x\neq 0$, one can use the operator
$i(\hat{\beta}_{\delta_{b}}^{(x)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}$
on $\mathcal{H}_{\mathfrak{b}}^{\tilde{\varepsilon}_{b}}$ with
$\tilde{\varepsilon}_{b}=\varepsilon_{b}+2x$, and thus has

$i(\hat{\beta}_{\delta_{b}}^{(x)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}=\hat{\beta}_{\delta_{b}}^{(0)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}.$
(34)

Therefore, without loss of generality, we can set $\mathfrak{b}=0$ and study
properties of
$\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
for any given $\varepsilon_{b}$.

From now on, let us refer to the restriction
$\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
for some $\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ as
$\hat{\beta}_{\delta_{b}}$ unless being specially noted. Because
$\hat{\beta}_{\delta_{b}}$ is unbounded, one has to assign certain domain to
complete its definition. A natural choice of the domain
$D(\hat{\beta}_{\delta_{b}})$ reads

$D(\hat{\beta}_{\delta_{b}})=\\{\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})},|\mathrm{supp}(\psi)|<\infty\\}$
(35)

where $|\mathrm{supp}(\psi)|$ denotes the cardinality of the support of
$\psi$. This implies that $D(\hat{\beta}_{\delta_{b}})$ only consists of
finite linear combination of the basis
$|\mu\rangle\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$. The essential
self-adjointness of $\hat{\beta}_{\delta_{b}}$ with the domain
$D(\hat{\beta}_{\delta_{b}})$ can be proven. Define a self-adjoint operator
$\hat{N}(\geq 1)$ as $\hat{N}|\mu\rangle=(1+\mu^{2})|\mu\rangle$. Then it is
straight-forward to verify that there exist numbers $c,d\in\mathbb{R}$ such
that

$\displaystyle\|\hat{\beta}_{\delta_{b}}\psi\|$ $\displaystyle\leq
c\|N\psi\|,\forall\psi\in D$ (36)
$\displaystyle\left|\langle\hat{\beta}_{\delta_{b}}\psi,\hat{N}\psi\rangle-\langle\hat{N}\psi,\hat{\beta}_{\delta_{b}}\psi\rangle\right|$
$\displaystyle\leq d\|\hat{N}^{1/2}\psi\|^{2},\ \forall\psi\in D.$

where $\langle\cdot,\cdot\rangle$ denotes the inner product in
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$. Therefore
$\hat{\beta}_{\delta_{b}}$ is essentially self-adjoint with the domain
$D(\hat{\beta}_{\delta_{b}})$ according to Theorem X.37 inReed (2003).

By expanding the eigen-equation
$\hat{\beta}_{\delta_{b}}|\psi\rangle=\omega|\psi\rangle$ with the basis
$|\mu\rangle$, one has

$\omega\psi(\mu)=\frac{1}{2i}\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}\Big{(}(\mu+2\delta_{b})\psi(\mu+2\delta_{b})-\mu\psi(\mu-2\delta_{b})\Big{)}.$
(37)

As shown in appendix A, $\psi(\mu)$ behaves asymptotically as

$\psi(\mu)=\left\\{\begin{aligned}
&\frac{1}{\sqrt{|\mu|}}\left(\chi_{+}e^{ik\ln(|\mu|)}+\chi_{-}e^{-ik\ln(|\mu|)}\right)+O(|\mu|^{-1}),\
\mu=\varepsilon_{b}+4n\delta_{b}\\\
&\frac{1}{\sqrt{|\mu|}}\left(\chi_{+}e^{ik\ln(|\mu|)}-\chi_{-}e^{-ik\ln(|\mu|)}\right)+O(|\mu|^{-1}),\
\mu=\varepsilon_{b}+(4n+2)\delta_{b}\end{aligned}\right.$ (38)

where $k=\frac{\omega}{\gamma\ell_{\mathrm{p}}^{2}\delta_{b}}$ and
$n\in\mathbb{Z}$. This implies that $\int|\psi(\mu)|^{2}{\rm d}\mu=\infty.$
and hence

$\sum_{\mu=\varepsilon_{b}+2n}|\psi(\mu)|^{2}=\infty.$

Thus $\psi(\mu)$ is unnormalizable. Moreover, since $\psi(\mu)$ behaves
asymptoticly as a plane wave of $\ln(|\mu|)$, we can show by using the same
techniques as in Zhang et al. (2018) and the Weyl’s criterion Reed (2003),
that the spectrum of $\widehat{p_{b}\sin(\delta_{b}b)}$ is the entire real
line $\mathbb{R}$. Furthermore, the two leading-order functions in (38),
denoted by

$\psi_{0}^{\pm}(\mu):=\frac{1}{\sqrt{|\mu|}}\left(\chi_{+}e^{ik\ln(|\mu|)}\pm\chi_{-}e^{-ik\ln(|\mu|)}\right),$

satisfy

$-i\gamma\ell_{\mathrm{p}}^{2}\delta_{b}\mathrm{sgn}(\mu)\sqrt{|\mu|}\frac{{\rm
d}}{{\rm d}\mu}\sqrt{|\mu|}\psi_{0}^{\pm}(\mu)=\omega\psi_{0}^{\mp}(\mu),$
(39)

where $\mathrm{sgn}(\mu)$ denotes the sign function of $\mu$. This implies
that the operator $\hat{\beta}_{\delta_{b}}$, in the large $\mu$ limit,
returns to the Schödinger quantization of $p_{b}b$,

$\widehat{p_{b}b}=-i\gamma\ell_{\mathrm{p}}^{2}\mathrm{sgn}(\mu)\sqrt{|\mu|}\frac{{\rm
d}}{{\rm d}\mu}\sqrt{|\mu|}.$ (40)

Therefore, the classical limit of $\hat{\beta}_{\delta_{b}}$ is correct. This
finishes the quantization procedure of $p_{b}\sin(\delta_{b}b)$ and the study
of the properties of the corresponding operator $\hat{\beta}_{\delta_{b}}$.
The same discussion can be transported analogously to $p_{c}\sin(\delta_{c}c)$
for constant $\delta_{c}$. The resulting operator
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ is self-adjoint and possesses
the entire line as its spectrum. These properties of
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ are compatible with our general
assumptions.

###  III.3 The operator $\hat{\mathfrak{h}}$

Let us denote the operator in the RHS of (25) as

$\hat{\mathfrak{h}}^{(m)}:=\frac{2L_{o}\gamma
m}{\delta_{b}}\hat{\beta}_{\delta_{b}}+\frac{1}{\delta_{b}^{2}}\hat{\beta}_{\delta_{b}}^{2}+\gamma^{2}\hat{p}_{b}^{2}.$
(41)

It should be noted that the operator $\hat{\beta}_{\delta_{b}}$ in (41) is the
original one defined by (30) with non-vanishing $\mathfrak{b}$. However, as
discussed below (33), the identity map $i$ from
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ to the specific Hilbert space
$\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}$ with
$\tilde{\varepsilon}_{b}=\varepsilon_{b}+2\mathfrak{b}$ ensures that we can
set $\mathfrak{b}=0$. Thus, by this treatment we define an operator in
$\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}$ corresponding to
$\hat{\mathfrak{h}}^{(m)}$ as

$i(\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}=\frac{2L_{o}\gamma
m}{\delta_{b}}i(\hat{\beta}_{\delta_{b}}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}+\frac{1}{\delta_{b}^{2}}i(\hat{\beta}_{\delta_{b}}^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}+\gamma^{2}i(\hat{p}_{b}^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}.$
(42)

where the map $i$, defined by (32), identifies the Hilbert spaces
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ and
$\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}$ with
$\tilde{\varepsilon}_{b}=\varepsilon_{b}+2\mathfrak{b}$. For the first two
terms in this operator, one has

$i((\hat{\beta}_{\delta_{b}})^{n}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}=(\hat{\beta}_{\delta_{b}}^{(0)})^{n}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}.$
(43)

where $\hat{\beta}_{\delta_{b}}^{(0)}$ is defined by (34). For the last term,
we have

$i(\hat{p}_{b}^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}|\tilde{\varepsilon}_{b}+2n\delta_{b}\rangle=[(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}]|\tilde{\varepsilon}_{b}+2n\delta_{b}\rangle.$
(44)

Therefore, (42) can be expressed as

$i(\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})})i^{-1}=\frac{2L_{o}\gamma
m}{\delta_{b}}(\hat{\beta}_{\delta_{b}}^{(0)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})})+\frac{1}{\delta_{b}^{2}}[(\hat{\beta}_{\delta_{b}}^{(0)})^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}]+\gamma^{2}[(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}\restriction\mathcal{H}_{\mathfrak{b}}^{(\tilde{\varepsilon}_{b})}].$
(45)

That is, by setting $\mathfrak{b}=0$ for the $\hat{\beta}_{\delta_{b}}$ in
$\hat{\mathfrak{h}}^{(m)}$, $\hat{p}_{b}$ has to be replaced by
$\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b}$. Thus we re-define
$\hat{\mathfrak{h}}^{(m)}$ as

$\displaystyle\hat{\mathfrak{h}}^{(m)}=$ $\displaystyle\frac{2L_{o}\gamma
m}{\delta_{b}}\hat{\beta}_{\delta_{b}}+\frac{1}{\delta_{b}^{2}}\hat{\beta}_{\delta_{b}}^{2}+\gamma^{2}(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2},$
(46)

where $\hat{\beta}_{\delta_{b}}$ is defined by (30) with $\mathfrak{b}=0$. As
a consequence, the operator $\hat{\mathfrak{h}}$ is changed correspondingly to

$(\hat{\mathfrak{h}}\psi)(m)=\hat{\mathfrak{h}}^{(m)}\psi(m)$ (47)

with the re-defined $\hat{\mathfrak{h}}^{(m)}$.

Now let us construct the separable Hilbert $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot))$ as mentioned below
(25). To do this, we only need to assign to each $m$ the Hilbert space
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m))}$, i.e., to define
$\mathcal{H}_{\mathfrak{b}}(\cdot)$ as

$\mathcal{H}_{\mathfrak{b}}(\cdot):m\mapsto\mathcal{H}_{\mathfrak{b}}(m):=\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m))},$
(48)

where $\varepsilon_{b}:m\mapsto\varepsilon_{b}(m)\in[0,2\delta_{b})$ is
assumed to be some sufficiently well-behaved function. Then the resulting
Hilbert space $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot))$ is supposed to be
acted by $\hat{\mathfrak{h}}$. To define the domain of the unbounded operator
$\hat{\mathfrak{h}}$, we first define the domain $D_{\mathfrak{b}}(m)$ of
$\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m))}$
for each $m$ by

$D_{\mathfrak{b}}(m):=\\{\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m))},\langle\mu|\psi\rangle\neq
0\text{ for finite numbers of }\mu\\}.$ (49)

Then the domain of $\hat{\mathfrak{h}}$, denoted by $D(\hat{\mathfrak{h}})$,
reads

$D(\hat{\mathfrak{h}})=\\{\psi\in L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot)),\psi(m)\in
D_{\mathfrak{b}}(m)\ \forall m\\}.$ (50)

We now complete the rigorous definition of the operator $\hat{\mathfrak{h}}$.
Since $\hat{\mathfrak{h}}$ is identical to the operator-valued function
$\hat{\mathfrak{h}}^{(\cdot)}:m\mapsto\hat{\mathfrak{h}}^{(m)}$, instead of
investigating $\hat{\mathfrak{h}}$ itself, it is sufficient to study
$\hat{\mathfrak{h}}^{(m)}$ for all $m$. In the following, to simplify our
notions, we will abbreviate $\varepsilon_{b}(m)$ to $\varepsilon_{b}$ if the
dependence of $\varepsilon_{b}$ on $m$ does not matter. Moreover, the
restricted operator
$\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
with the domain $D_{\mathfrak{b}}(m)$ is denoted simply by
$\hat{\mathfrak{h}}^{(m)}$. Furthermore, when $\hat{\mathfrak{h}}$ is
mentioned, it refers to the operator $\hat{\mathfrak{h}}$ with domain
$D(\hat{\mathfrak{h}})$.

The operator $\hat{\mathfrak{h}}$ and $\hat{\mathfrak{h}}^{(m)}$ are necessary
to be self-adjoint to govern a well-defined dynamics. By using the Kato-
Rellich Theorem Reed et al. (2003a), we can prove that both
$\hat{\mathfrak{h}}$ and $\hat{\mathfrak{h}}^{(m)}$ are essentially self-
adjoint (see Appendix B for details). Because of the essential self-
adjointness of $\hat{\mathfrak{h}}$ and $\hat{\mathfrak{h}}^{(m)}$, their
closure, denoted by $\overline{\hat{\mathfrak{h}}^{(m)}}$ and
$\overline{\hat{\mathfrak{h}}}$ respectively, can be regarded as the
Hamiltonian operator of the current model with the desired properties. We use
$\overline{D_{\mathfrak{b}}(m)}$ and $\overline{D(\hat{\mathfrak{h}})}$ to
denote the domains of $\overline{\hat{\mathfrak{h}}^{(m)}}$ and
$\overline{\hat{\mathfrak{h}}}$ respectively.

##  IV Dynamics govern by $\overline{\hat{\mathfrak{h}}}$

To solve the dynamics governed by $\overline{\hat{\mathfrak{h}}}$, one needs
to diagonalize $\overline{\hat{\mathfrak{h}}}$, or equivalently, to
diagonalize $\overline{\hat{\mathfrak{h}}^{(m)}}$ for all $m$. To begin with,
we first study the discreteness of the spectrum of
$\overline{\hat{\mathfrak{h}}^{(m)}}$. By definition,
$\hat{\mathfrak{h}}^{(m)}$ can be re-expressed as

$\hat{\mathfrak{h}}^{(m)}=-L_{0}^{2}m^{2}\gamma^{2}+\left(L_{0}m\gamma+\frac{1}{\delta_{b}}\hat{\beta}_{\delta_{b}}\right)^{2}+\gamma^{2}(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}.$
(51)

Thus, one has

$\langle\psi|\overline{\hat{\mathfrak{h}}^{(m)}}|\psi\rangle\geq\gamma^{2}\langle\psi|(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}|\psi\rangle-
L_{0}^{2}m^{2}\gamma^{2}\langle\psi|\psi\rangle,\
\forall|\psi\rangle\in\overline{D_{\mathfrak{b}}(m)}.$ (52)

Hence $\overline{\hat{\mathfrak{h}}^{(m)}}$ is bounded from below. By (52), we
can obtain

$\mu_{n}(\overline{\hat{\mathfrak{h}}^{(m)}})\geq-
L_{0}^{2}m^{2}\gamma^{2}+\gamma^{2}\mu_{n}((\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2})$
(53)

where, for an operator $\hat{A}$, $\mu_{n}(\hat{A})$ denotes

$\mu_{n}(\hat{A}):=\sup_{\varphi_{1},\cdots,\varphi_{n-1}\in\mathcal{H}_{\mathfrak{b}}^{\varepsilon_{b}}}\inf_{\begin{subarray}{c}\psi\in\overline{D_{\mathfrak{b}}(m)};\|\psi\|=1;\\\
\langle\varphi_{i},\psi\rangle=0,\forall
i=1,\cdots,n-1\end{subarray}}\langle\psi,\hat{A}\psi\rangle.$ (54)

Note that $(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}$ can be
defined on $\overline{D_{\mathfrak{b}}(m)}$ by definition. Hence
$\mu_{n}((\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2})$ is well-
defined. Since
$\mu_{n}((\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2})\to\infty$
as $n\to\infty$, one has that

$\lim_{n\to\infty}\mu_{n}(\overline{\hat{\mathfrak{h}}^{(m)}})=\infty.$ (55)

Hence, according to the min-max principle (see, e.g., Theorem XIII.1 in Reed
et al. (2003b)), $\overline{\hat{\mathfrak{h}}^{(m)}}$ has purely discrete
spectrum. In other words, each element in the spectrum of
$\overline{\hat{\mathfrak{h}}^{(m)}}$, denoted by
$\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})$, is an eigenvalue of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ with finite multiplicity.

Given the significance of $\overline{\hat{\mathfrak{h}}^{(m)}}$, it is
desirable to understand the properties of $\sigma(\hat{\mathfrak{h}}^{(m)})$.
In particular, one may ask whether the eigenvalue $\omega(m)$ as a function of
$m$ is analytic or not. This issue is closely related to the analyticity of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ on $m$ in the sense of Kato Reed et al.
(2003b); Kato (2013). To overcome the technical difficulty that
$\overline{\hat{\mathfrak{h}}^{(m)}}$ for different $m$ are defined in
different Hilbert spaces, we employ the following unitary map for a given
$m_{o}$,

$\mathfrak{i}_{m}:\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m))}\ni|\varepsilon_{b}(m)+2n\delta_{b}^{(m)}\rangle\mapsto|\varepsilon_{b}(m_{o})+2n\delta_{b}^{(m_{o})}\rangle\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b}(m_{o}))}$
(56)

where the dependence of $\varepsilon_{b}$ and $\delta_{b}$ on $m$ is written
explicitly. The issue on the analyticity of $\omega(m)$ can be equivalently
discussed by that of
$\mathfrak{i}_{m}\overline{\hat{\mathfrak{h}}^{(m)}}\mathfrak{i}_{m}^{-1}$
which is defined on
$\mathfrak{i}_{m}\overline{D_{\mathfrak{b}}(m)}=\overline{D_{\mathfrak{b}}(m_{o})}$.
For simplicity, let us use $\delta_{b}^{o}$ and $\varepsilon_{b}^{o}$ to
denote $\delta_{b}^{(m_{o})}$ and $\varepsilon_{b}(m_{o})$ respectively.

An sesquilinear form $T_{m}(\cdot,\cdot)$ associated to
$\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}$ can be defined
on $D_{\mathfrak{b}}(m_{o})$ by

$T_{m}(\psi,\phi)=\langle\psi|\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}|\phi\rangle,\
\psi,\phi\in D_{\mathfrak{b}}(m_{o}).$ (57)

Then (52) implies that $T_{m}$ is semibounded, i.e.,

$T_{m}(\psi,\psi)\geq-L_{0}^{2}m^{2}\gamma^{2}\|\psi\|^{2},\ \forall\psi\in
D_{\mathfrak{b}}(m_{o}).$ (58)

Hence, $T_{m}$ is closable. Its closure, denoted by $\overline{T_{m}}$, is the
extension of $T_{m}$ on the closure of $D_{\mathfrak{b}}(m_{o})$ with respect
to the norm

$\|\psi\|_{m}=\sqrt{T_{m}(\psi,\psi)+(L_{0}^{2}m^{2}\gamma^{2}+1)\langle\psi|\psi\rangle}.$
(59)

Since the norm $\|\psi\|_{m}$ depends on $m$, the resulting closures $Q(m)$ of
$D_{\mathfrak{b}}(m_{o})$ could depend on $m$ in general. Suppose that
$\delta_{b}^{(m)}$ and $\varepsilon_{b}(m)$ are analytic functions of $m$ at
$m_{o}$ and $\delta_{b}^{(m_{o})}\neq 0$. It turns out that $\overline{T_{m}}$
is an analytic family of type (a) in the sense of Kato (see section VII.4 in
Kato (2013)). This, by definition, is ensured by that i) $Q(m)=Q(m_{o})$ for
all $m$ sufficiently close to $m_{o}$, and ii) $\overline{T_{m}}(\psi,\psi)$
is analytic on $m$ for all $\psi\in Q(m_{o})$. The detailed proof of this
conclusion is presented in Appendix C.

Because $\overline{T_{m}}$ is symmetric and closed, there is a self-adjoint
operator $\hat{\mathfrak{t}}_{m}$ associated to it, which is indeed the
Friedrichs extension of $\hat{\mathfrak{h}}^{(m)}$. By the analyticity of
$\overline{T_{m}}$, $\hat{\mathfrak{t}}_{m}$ forms an analytic family of type
(B) in the sense of Kato (see Section VII.4 in Kato (2013)). As a consequence,
$\mathfrak{i}_{m}\overline{\hat{\mathfrak{h}}^{(m)}}\mathfrak{i}_{m}^{-1}$ and
thus $\overline{\hat{\mathfrak{h}}^{(m)}}$ carry the same property because of
$\hat{\mathfrak{t}}_{m}=\mathfrak{i}_{m}\overline{\hat{\mathfrak{h}}^{(m)}}\mathfrak{i}_{m}^{-1}$,
which can be proven as follows. Firstly, the domain
$\overline{D_{\mathfrak{b}}(m_{o})}$ of
$\mathfrak{i}_{m}\overline{\hat{\mathfrak{h}}^{(m)}}\mathfrak{i}_{m}^{-1}$ is
the closure of $D_{\mathfrak{b}}(m_{o})$ with respect to the graph norm

$\|\psi\|_{g}:=\sqrt{\|\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}\psi\|^{2}+\|\psi\|^{2}}.$

Secondly, applying the same techniques in the proof of Lemma C.1 in Appendix
C, one can show straight-forwardly that $\|\cdot\|_{g}$ is equivalent to the
norm $\|\cdot\|_{g}^{\prime}$ defined by

$\|\psi\|_{g}^{\prime}:=\sqrt{\langle\psi|\hat{p}_{b}^{4}|\psi\rangle+\langle\psi|\psi\rangle}.$

Finally, because of $\|\psi\|_{+}\leq\|\psi\|_{g}^{\prime}$ for all $\psi\in
D_{\mathfrak{b}}(m_{o})$, one concludes
$\overline{D_{\mathfrak{b}}(m_{o})}\subset Q(m)$, which results in
$\hat{\mathfrak{t}}_{m}=\mathfrak{i}_{m}\overline{\hat{\mathfrak{h}}^{(m)}}\mathfrak{i}_{m}^{-1}$
according to the uniqueness of the Friedrichs extension.

Due to the analyticity of $\overline{\hat{\mathfrak{h}}^{(m)}}$, the
analyticity of $\omega(m)$ can obtained directly (see e.g. Chapter XII of Reed
et al. (2003b) and Theorem VII.1.8 in Kato (2013)), which is summarized
precisely as the following theorem.

######  Theorem IV.1.

Suppose that $\delta_{b}^{(m)}$ and $\varepsilon_{b}(m)$ are analytic
functions of $m$ at $\mathring{m}$ and $\delta_{b}^{(\mathring{m})}\neq 0$.
Given an eigenvalue $\mathring{\omega}$ of
$\overline{\hat{\mathfrak{h}}^{(\mathring{m})}}$ which is of an algebraic
multiplicity $k$, for each $m$ which is sufficiently close to $\mathring{m}$,
$\overline{\hat{\mathfrak{h}}^{(m)}}$ has exactly $k$ eigenvalues (counting
multiplicity) near $\mathring{\omega}$. These eigenvalues are given by $p\
(\leq k)$ distinct, single-valued and analytic functions
$\omega_{1}(m),\cdots,\omega_{p}(m)$.

While this theorem is valid for the general cases that the eigenvalue $\omega$
possesses the algebraic multiplicity $k\geq 1$, it can be seen from the
numerical results in the next section that each eigenspace of
$\hat{\mathfrak{h}}^{(m)}$ for all $m$ is exactly one-dimensional.

###  IV.1 Numerical approach to compute the eigenvalues and eigenstates

Since $\overline{\hat{\mathfrak{h}}^{(m)}}$ has only discrete spectrum, the
eigenvector $|\psi\rangle$ associated to each
$\omega\in\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})$ is normalizable. Hence
the function $\psi(\mu):=\langle\mu|\psi\rangle$ decreases rapidly for
sufficiently large values of $|\mu|$. This fact motivates us to use the
finite-dimensional cut-off approximation to collect eigenvalues of
$\overline{\hat{\mathfrak{h}}^{(m)}}$. More precisely, we consider a finitely
dimensional subspace, denoted by
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}$, spanned by
$|\mu_{\eta}\rangle=|\varepsilon_{b}+2\eta\delta_{b}\rangle\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$
with $|\eta|\leq k$ for some large $k$. Let $\hat{P}^{(k)}$ be the projection
operator to $\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}$ such that

$\hat{P}^{(k)}|\mu_{\eta}\rangle=\left\\{\begin{array}[]{cc}|\mu_{\eta}\rangle,&|\eta|\leq
k,\\\ 0,&\text{otherwise}.\end{array}\right.$ (60)

Given an eigenvector $|\psi\rangle$ of $\overline{\hat{\mathfrak{h}}^{(m)}}$
with respect to an eigenvalue $\omega$, we have

$\displaystyle(\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}-\omega)|\psi\rangle=(\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}-\overline{\hat{\mathfrak{h}}^{(m)}})|\psi\rangle$
(61) $\displaystyle=$
$\displaystyle-(1-\hat{P}^{(k)})\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}|\psi\rangle-\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}(1-\hat{P}^{(k)})|\psi\rangle-(1-\hat{P}^{(k)})\overline{\hat{\mathfrak{h}}^{(m)}}(1-\hat{P}^{(k)})|\psi\rangle.$

Since $\psi(\mu):=\langle\mu|\psi\rangle$ rapidly decreases for sufficiently
large values of $|\mu|$, the term $1-\hat{P}^{(k)}$ in the RHS of (61)
indicates that
$(\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}-\omega)|\psi\rangle$
should be very small for large $k$. Thus it is reasonable to expect that
$\omega$ and $|\psi\rangle$ can be approximated by certain eigenvalue and its
corresponding eigenvector of
$\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$ respectively
for large $k$. This is the reason for our finite-dimensional cut-off
approximation method. To apply this method, we need to identify those
eigenvalues of $\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$
suitable for approximating the eigenvalues of $\hat{\mathfrak{h}}^{(m)}$, and
check whether all eigenvalues of $\hat{\mathfrak{h}}^{(m)}$ can be
approximated by this way.

Since $\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$ is a
$(2k+1)$ dimensional symmetric matrix under the basis $|\mu\rangle$, it has
$(2k+1)$ eigenvalues. We denote them as $\lambda_{i}^{(k)}$ with
$\lambda_{1}^{(k)}\leq\lambda_{2}^{(k)}\leq\cdots\leq\lambda_{2k+1}^{(k)}$,
where the subscript $i$ denoted that $\lambda_{i}^{(k)}$ is the $i$th
eigenvalue from the least and the superscript $k$ corresponds to the
superscript of $\hat{P}^{(k)}$. Obviously, we have $i\leq 2k+1$ in
$\lambda_{i}^{(k)}$. Thus, once the $i$th eigenvalue of
$\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$ for some $k$
is mentioned, $k$ should satisfy $k\geq k_{o}(i)$ with $k_{o}(i):=(i-1)/2$.
Moreover, because $\overline{\hat{\mathfrak{h}}^{(m)}}$ is semibounded, it has
the minimal eigenvalue. Thus its eigenvalues can be denoted by $\omega_{i}$
with $i=1,2,\cdots$ such that
$\omega_{1}\leq\omega_{2}\leq\cdots\leq\omega_{n}\leq\cdots$. Then by using
the Rayleigh-Reitz technique (see, e.g., Theorem XIII.3 in Reed et al. (2003b)
and Appendix D), one can obtain

$\lambda_{i}^{(k_{o}(i))}\geq\lambda_{i}^{(k_{o}(i)+1)}\geq\cdots\geq\lambda_{i}^{(k_{o}(i)+n)}\geq\cdots\geq\omega_{i}.$
(62)

As a consequence, the limit of $\lambda_{i}^{(k)}$ as $k\to\infty$ exists,
i.e.,

$\lambda_{i}:=\lim_{k\to\infty}\lambda_{i}^{(k)}<\infty.$ (63)

Consider the sequence
$\\{\psi_{i}^{(k_{o}(i))},\psi_{i}^{(k_{o}(i)+1)},\cdots,\psi_{i}^{(k_{o}(i)+n)},\cdots\\}$,
where, for each $k$, $\psi_{i}^{(k)}$ is an eigenvector of
$\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)}$ with the eigenvalue
$\lambda_{i}^{(k)}$ and satisfies $\|\psi_{i}^{(k)}\|=1$. Because of
$\|\psi_{k}^{(k)}\|=1$, the sequence
$\\{\psi_{i}^{(k_{o}(i))},\psi_{i}^{(k_{o}(i)+1)},\cdots,\psi_{i}^{(k_{o}(i)+n)},\cdots\\}$
contains a subsequence by the Banach-Alaoglu theorem, i.e., there exists
$\psi_{i}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}$ such that

$\lim_{l\to\infty}\langle\psi_{i}^{(n_{l})}|\varphi\rangle=\langle\psi_{i}|\varphi\rangle,\
\forall\varphi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}.$ (64)

Consider all weakly convergent subsequences of
$\\{\psi_{i}^{(k_{o}(i))},\psi_{i}^{(k_{o}(i)+1)},\cdots,\psi_{i}^{(k_{o}(i)+n)},\cdots\\}$
and collect their limits defined by (64). Denote the space spanned by these
limits as $\Lambda_{i}$. It turns out that each $\lambda_{i}$ is an eigenvalue
of $\overline{\hat{\mathfrak{h}}^{(m)}}$ and all the elements in $\Lambda_{i}$
are eigenvectors corresponding to the eigenvalue $\lambda_{i}$ (see Theorem
E.1 in Appendix E for proof). Therefore, the $i$th eigenvalue and its
corresponding eigenvectors of
$\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$ with $k\gg i$
approximate some eigenvalue and eigenvectors of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ respectively. To check whether all
eigenvalues of $\hat{\mathfrak{h}}^{(m)}$ can be approximated by the finite-
dimensional cut-off approximation, we can show that

$\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})\cap(\lambda_{i},\lambda_{i+1})=\emptyset,$
(65)

if $\lambda_{i}\neq\lambda_{i+1}$ (see Theorem E.2 in Appendix E for more
details). In other words, each eigenvalue of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ is a limit point of the sequence
$\\{\lambda_{j}^{k_{o}(j)},\lambda_{j}^{k_{o}(j)+1},\cdots,\lambda_{j}^{k_{o}(j)+n},\cdots\\}$
for some $j$.

The accuracy of the above approximation can be discussed by the following
procedure as in Kato (1949); Harrell (1978). Let $|\psi_{i}^{(k)}\rangle$ be a
normalized eigenvector of
$\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}$ corresponding
to the eigenvalue $\lambda_{i}^{(k)}$. One has

$\langle\psi_{i}^{(k)}|\overline{\hat{\mathfrak{h}}^{(m)}}|\psi_{i}^{(k)}\rangle=\lambda_{k}^{(k)}.$
(66)

Defining

$\epsilon_{i}^{(k)}:=\|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i}^{(k)})\psi_{i}^{(k)}\|,$
(67)

we will show

$\lambda_{i}\in(\lambda_{i}^{(k)}-\epsilon_{i}^{(k)},\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}).$
(68)

Let $\\{|\omega,\delta\rangle\\}$ be an orthonormal basis of the Hilbert
space, consisting of eigenvectors of $\overline{\hat{\mathfrak{h}}^{(m)}}$,
where $\omega$ denotes the eigenvalue and $\delta$ represents other quantum
numbers. Given $\lambda_{i}^{(k)}$, for arbitrary real numbers $\alpha$ and
$\beta$ with $\alpha<\lambda_{i}^{(k)}<\beta$, we have

$\langle\psi_{i}^{(k)}|(\overline{\hat{\mathfrak{h}}^{(m)}}-\alpha)(\overline{\hat{\mathfrak{h}}^{(m)}}-\beta)|\psi_{i}^{(k)}\rangle=\sum_{\omega}\sum_{\delta}(\omega-\alpha)(\omega-\beta)|\langle\psi_{i}^{(k)}|\omega,\delta\rangle|^{2}.$
(69)

Then supposing
$(\alpha,\beta)\cap\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})=\emptyset$, one
will get

$\langle\psi_{i}^{(k)}|(\overline{\hat{\mathfrak{h}}^{(m)}}-\alpha)(\overline{\hat{\mathfrak{h}}^{(m)}}-\beta)|\psi_{i}^{(k)}\rangle\geq
0.$ (70)

Because of
$\langle\psi_{i}^{(k)}|(\hat{\mathfrak{h}}^{(m)}-\lambda_{i}^{(k)})|\psi_{i}^{(k)}\rangle=0$,
$\langle\psi_{i}^{(k)}|(\overline{\hat{\mathfrak{h}}^{(m)}}-\alpha)(\overline{\hat{\mathfrak{h}}^{(m)}}-\beta)|\psi_{i}^{(k)}\rangle$
can be expanded as

$\langle\psi_{i}^{(k)}|(\overline{\hat{\mathfrak{h}}^{(m)}}-\alpha)(\overline{\hat{\mathfrak{h}}^{(m)}}-\beta)|\psi_{i}^{(k)}\rangle=(\epsilon_{i}^{(k)})^{2}+(\lambda_{i}^{(k)}-\alpha)(\lambda_{i}^{(k)}-\beta).$
(71)

Substituting (71) into (70), we get

$\beta\leq\lambda_{i}^{(k)}+\frac{(\epsilon_{i}^{(k)})^{2}}{\lambda_{i}^{(k)}-\alpha}.$
(72)

Note that (72) holds under the assumption
$(\alpha,\beta)\cap\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})=\emptyset$.
Thus, if
$\beta>\lambda_{i}^{(k)}+(\epsilon_{i}^{(k)})^{2}/(\lambda_{i}^{(k)}-\alpha)$,
one will get
$(\alpha,\beta)\cap\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})\neq\emptyset$,
which, together with the fact that
$\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})$ is closed, ensures

$\left(\alpha,\lambda_{i}^{(k)}+\frac{(\epsilon_{i}^{(k)})^{2}}{\lambda_{i}^{(k)}-\alpha}\right]\cap\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})\neq\emptyset.$
(73)

Choosing $\alpha$ as $\alpha=\lambda_{i}^{(k)}-\epsilon_{i}^{(k)}$, we get

$\left(\lambda_{i}^{(k)}-\epsilon_{i}^{(k)},\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}\right]\cap\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})\neq\emptyset.$
(74)

Noting that (74) is true for all $k\geq k_{o}(i)$ and
$\lim_{k\to\infty}\lambda_{i}^{(k)}\pm\epsilon_{i}^{(k)}=\lambda_{i}$, one
finally has

$\lambda_{i}\in(\lambda_{i}^{(k)}-\epsilon_{i}^{(k)},\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}].$
(75)

Similar to the derivation of (72), one can also obtain

$\alpha\geq\lambda_{i}^{(k)}+\frac{(\epsilon_{i}^{(k)})^{2}}{\lambda_{i}^{(k)}-\beta},$
(76)

By choosing $\beta=\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}$, (76) will finally
lead to

$\lambda_{i}\in[\lambda_{i}^{(k)}-\epsilon_{i}^{(k)},\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}).$
(77)

Then (68) is obtained by combining (75) and (77).

The above discussion can also be applied to estimate the accuracy of
approximating $|\psi_{i}\rangle$ by $|\psi_{i}^{(k)}\rangle$. Given an
interval $(\alpha_{o},\beta_{o})\ni\lambda_{i}$ such that $\lambda_{i}$ is the
only eigenvalue contained in it, we have

$\displaystyle\langle\psi_{i}^{(k)}|(\hat{\mathfrak{h}}^{(m)}-\alpha_{o})(\hat{\mathfrak{h}}^{(m)}-\beta_{o})|\psi_{i}^{(k)}\rangle-(\lambda_{i}-\alpha_{o})(\lambda_{i}-\beta_{o})\sum_{\delta}|\langle\psi_{i}^{(n)}|\lambda_{i},\delta\rangle|^{2}$
(78) $\displaystyle=$
$\displaystyle\sum_{\lambda\neq\lambda_{i}}\sum_{\delta}(\lambda-\alpha)(\lambda-\beta)\|\langle\psi_{k}^{(n)}|\lambda,\delta\rangle\|\geq
0.$

Substituting (71) into (78), one obtains

$(\epsilon_{i}^{(k)})^{2}+(\lambda_{i}^{(k)}-\alpha)(\lambda_{i}^{(k)}-\beta)\geq(\lambda_{i}-\alpha_{o})(\lambda_{i}-\beta_{o})\sum_{\delta}|\langle\psi_{i}^{(n)}|\lambda_{i},\delta\rangle|^{2}$
(79)

which leads to

$1-\sum_{\delta}\left|\langle\psi_{i}^{(k)}|\lambda_{i},\delta\rangle\right|^{2}\leq\frac{(\varepsilon_{i}^{(k)})^{2}}{(\lambda_{i}^{(k)}-\alpha_{o})(\beta_{o}-\lambda_{i}^{(k)})}.$
(80)

Now let us assume
$\lambda_{i-1}^{(k)}\neq\lambda_{i}^{(k)}\neq\lambda_{i+1}^{(k)}$ without loss
of generality. Then one can choose a sufficiently large $k$ such that

$\lambda_{i-1}^{(k)}+\epsilon_{i-1}^{(k)}<\lambda_{i}^{(k)}-\epsilon_{i}^{(k)},\
\lambda_{i}^{(k)}+\epsilon_{i}^{(k)}<\lambda_{i+1}^{(k)}-\epsilon_{i+1}^{(k)},$
(81)

which, together with (68), implies that the interval
$(\lambda_{i-1}^{(k)}+\epsilon_{i-1}^{(k)},\lambda_{i+1}^{(k)}-\epsilon_{i+1}^{(k)})$
contains the single eigenvalue $\lambda_{i}$. Then, according to (80), we get

$1-\sum_{\delta}\left|\langle\psi_{i}^{(k)}|\lambda_{i},\delta\rangle\right|^{2}\leq\frac{(\epsilon_{i}^{(k)})^{2}}{(\lambda_{i}^{(k)}-\lambda_{i-1}^{(k)}-\epsilon_{i-1}^{(k)})(\lambda_{i+1}^{(k)}-\lambda_{i}^{(k)}-\epsilon_{i+1}^{(k)})}.$
(82)

The left hand side of (82) measures the accuracy of approximating
$|\psi_{i}\rangle$ by $|\psi_{i}^{(k)}\rangle$ because
$|\psi_{i}^{(k)}\rangle$ is normalized.

###  IV.2 Finite-dimensional cut-off approximation for the numerical
computation

In our model, there are two free parameters $\varepsilon_{b}$ and
$\mathfrak{b}$. By mimicking the derivation of Theorem C.1 in Appendix C, one
can show that $\overline{\hat{\mathfrak{h}}^{(m)}}$, as an operator-valued
function of $\varepsilon_{b}$ and $\mathfrak{b}$, form an analytic family of
type (B) in the sense of Kato. Therefore, we can set
$\varepsilon_{b}=0=\mathfrak{b}$ in our computation, and the results for case
with either $\varepsilon_{b}\neq 0$ or $\mathfrak{b}\neq 0$ can be obtained by
perturbing that for the case of $\varepsilon_{b}=0=\mathfrak{b}$. In other
words, the former can be expanded as some power series of $\varepsilon_{b}$
and $\mathfrak{b}$, and the convergences of the series are ensured by the
analyticity of $\hat{\mathfrak{h}}^{(m)}$ on $\varepsilon_{b}$ and
$\mathfrak{b}$.

By setting $\varepsilon_{b}=0$, we work in the specific Hilbert space define
by (31) with $\varepsilon_{b}=0$. This Hilbert space is denoted by
$\mathcal{H}^{(0)}$ where a state is given by a wave function
$\psi(2\eta\delta_{b})\equiv\psi(\eta)$ with $\eta\in\mathbb{Z}$. Then the
action of $\hat{\mathfrak{h}}^{(m)}$ reads

$\displaystyle(\overline{\hat{\mathfrak{h}}^{(m)}}\psi)(\eta)=-\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}(\eta+2)(\eta+1)\psi(\eta+2)-i\gamma^{2}\ell_{\mathrm{p}}^{2}L_{0}m(\eta+1)\psi(\eta+1)$
(83)
$\displaystyle+\left(\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}(\eta+1)^{2}+\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}(1+4\delta_{b}^{2}\gamma^{2})\eta^{2}\right)\psi(\eta)+i\gamma^{2}\ell_{\mathrm{p}}^{2}L_{0}m\eta\psi(\eta-1)-\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}\eta(\eta-1)\psi(\eta-2).$

Thus the eigen-equation
$(\overline{\hat{\mathfrak{h}}^{(m)}}\psi)(\eta)=\omega\psi(\eta)$ results in

$\displaystyle\psi(\eta+2)=\frac{4}{\gamma^{2}\ell_{\mathrm{p}}^{4}(\eta+2)(\eta+1)}\Big{(}-\omega\psi(\eta)-i\gamma^{2}\ell_{\mathrm{p}}^{2}L_{0}m(\eta+1)\psi(\eta+1)$
(84)
$\displaystyle+\left(\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}(\eta+1)^{2}+\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}(1+4\delta_{b}^{2}\gamma^{2})\eta^{2}\right)\psi(\eta)+i\gamma^{2}\ell_{\mathrm{p}}^{2}L_{0}m\eta\psi(\eta-1)-\frac{1}{4}\gamma^{2}\ell_{\mathrm{p}}^{4}\eta(\eta-1)\psi(\eta-2)\Big{)}.$

For $\eta=0$, (84) becomes

$\psi(2)=\left(-\frac{2\omega}{\gamma^{2}\ell_{\mathrm{p}}^{4}}+\frac{1}{2}\right)\psi(0)-i\frac{2L_{0}m}{\ell_{\mathrm{p}}^{2}}\psi(1),$
(85)

which implies that the value of $\psi(2)$ depends only on $\psi(0)$ and
$\psi(1)$. Once $\psi(2)$ is obtained by (85), the values of $\psi(\eta)$ for
all $\eta\geq 3$ can be obtained by (84). Thus, the values of $\psi(\eta)$ for
all $\eta\geq 2$ are determined completely by $\psi(0)$ and $\psi(1)$ by
(84)555Note that the values of $\psi(0)$ and $\psi(1)$ have to be chosen
suitably so that the resulting $\psi(\eta)$ is normalizable. . Similarly, one
can rewrite (84) to express $\psi(\eta-2)$ in terms of $\psi(\eta-1)$,
$\psi(\eta)$, $\psi(\eta+1)$ and $\psi(\eta+2)$. Then, by setting $\eta=-1$,
it is easy to see that the values of $\psi(\eta)$ for all $\eta\leq-3$ are
completely determined by $\psi(-1)$ and $\psi(-2)$. Thus, for a given
eigenvector of $\overline{\hat{\mathfrak{h}}^{(m)}}$, its values for $\eta<0$
decouple from its values for $\eta\geq 0$. Hence, the eigenvectors of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ can be classified into two supper-
selected sectors. The first sector consists of those $\psi$ vanishing for
$\eta<0$, while the second sector consists of the ones vanishing for $\eta\geq
0$. Consider a transformation $T$ defined as

$(T\psi)(\eta)=\psi(-\eta-1).$ (86)

Then $T$ relates the eigenvectors in the two sectors. It should be noted that
this classification of the eigenvectors is valid only for the case of
$\varepsilon_{b}=0=\mathfrak{b}$. However, for the case of
$\varepsilon_{b}\neq 0$ or $\mathfrak{b}\neq 0$, the eigenvectors can be
divided into two sectors, satisfying $\sum_{\eta\geq 0}|\psi(\eta)|^{2}\ll 1$
or $\sum_{\eta<0}|\psi(\eta)|^{2}\ll 1$ respectively.

Given an eigenvector $\psi_{-}$ in the second sector with the eigenvalue
$\omega_{-}$, a straight-forward calculation gives that

$(\overline{\hat{\mathfrak{h}}^{(m)}}+\hat{\epsilon})T\psi_{-}=\omega_{-}T\psi_{-},$
(87)

where $\hat{\epsilon}$ satisfies

$(\hat{\epsilon}\psi)(\eta)=-\gamma^{4}\ell_{\mathrm{p}}^{4}\delta_{b}^{2}(2\eta+1)\psi(\eta).$
(88)

Thus $T\psi_{-}$ is an eigenvector of
$\overline{\hat{\mathfrak{h}}^{(m)}}+\hat{\epsilon}$ with respect to the
eigenvalue $\omega_{-}$.

Define
$\hat{H}(\lambda)=(\overline{\hat{\mathfrak{h}}^{(m)}}+\hat{\epsilon})+\lambda\hat{\epsilon}$
on $\overline{D_{\mathfrak{b}}(m)}$. Similar to the discussion on
$\overline{\hat{\mathfrak{h}}^{(m)}}$, we can show that $\hat{H}(\lambda)$ is
self-adjoint with the domain $\overline{D_{\mathfrak{b}}(m)}$ independent of
$\lambda$. Moreover, $\hat{H}(\lambda)$ forms an analytic family of type (A)
in the sense of Kato. This can be verified easily by showing that
$\hat{H}(\lambda)|\psi\rangle$ for all
$|\psi\rangle\in\overline{D_{\mathfrak{b}}(m)}$ is a vector-valued analytic
function of $\lambda$ (see e.g. Chapter XII.1 in Reed et al. (2003b)).
Therefore, the Rayleigh-Schrödinger perturbation theory can be applied to
expand the eigenvalues $\Omega^{(\lambda)}$ and the eigenvectors
$\Psi^{(\lambda)}$ of $\hat{H}(\lambda)$ as the Rayleigh-Schrödinger series

$\displaystyle\Omega^{(\lambda)}$
$\displaystyle=\omega_{-}+\sum_{n=1}^{\infty}c_{n}(\lambda\gamma^{4}\ell_{\mathrm{p}}^{4}\delta_{b}^{2})^{n},$
(89) $\displaystyle\Psi^{(\lambda)}$
$\displaystyle=T\psi_{-}+\sum_{k=1}^{\infty}(\lambda\gamma^{4}\ell_{\mathrm{p}}^{4}\delta_{b}^{2})^{n}\phi_{n},$

where $c_{n}$ and $\phi_{n}$ denotes the coefficients (see e.g.Reed et al.
(2003b)) and the convergence of these series for all $\lambda\in\mathbb{R}$ is
ensured by the analyticity of $\hat{H}(\lambda)$ on $\lambda$. By setting
$\lambda=-1$, one gets the eigenvalue $\Omega^{(-1)}=:\omega_{+}$ and the
corresponding eigenvector $\Psi^{(-1)}=:\psi_{+}$ of the operator
$\hat{H}(-1)=\hat{\mathfrak{h}}^{(m)}$. By substituting the explicit
expression of $\phi_{n}$ into (89), one can get $\psi_{+}(\eta)=0$ for all
$\eta<0$. Thus $\psi_{+}$ are in the first sector. Therefore, for each
eigenvector of $\hat{\mathfrak{h}}^{(m)}$ in the second sector with eigenvalue
$\omega_{-}$, there always exist an adjoint eigenvector $\psi_{+}$ of
$\hat{\mathfrak{h}}^{(m)}$ in the first section with eigenvalue $\omega_{+}$
nearby $\omega_{-}$.

By (89), for a very small value of
$\gamma^{4}\ell_{\mathrm{p}}^{4}\delta_{b}^{2}$, the difference between
$\omega_{+}$ and $\omega_{-}$ would be very tiny so that very high
computational cost is needed to separate their values numerically. To overcome
this difficulty we consider the Hilbert space
$\mathcal{H}_{\mathfrak{b}}^{(0+)}\subset\mathcal{H}_{\mathfrak{b}}^{(0)}$
defined by

$\mathcal{H}_{\mathfrak{b}}^{(0+)}=\overline{\\{\psi\in\mathcal{H}_{\mathfrak{b}}^{(0)},\psi(\eta)=0\
\forall\eta<0\\}}$ (90)

where the symbol $\overline{\\{\cdot\\}}$ represents the completion with
respect to the inner product of $\mathcal{H}_{\mathfrak{b}}^{(0)}$. Then, we
diagonalize the operators
$\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(0+)}$ and
$(\hat{\mathfrak{h}}^{(m)}+\hat{\epsilon})\restriction\mathcal{H}_{\mathfrak{b}}^{(0+)}$,
denoting the restrictions of $\hat{\mathfrak{h}}^{(m)}$ and
$\hat{\mathfrak{h}}^{(m)}+\hat{\epsilon}$ on
$\mathcal{H}_{\mathfrak{b}}^{(0+)}$ respectively, by the finite-dimensional
cut-off approximation method. Let $\psi_{+}$ be an eigenvector of
$\hat{\mathfrak{h}}^{(m)}\restriction\mathcal{H}_{\mathfrak{b}}^{(0+)}$ with
eigenvalue $\omega_{+}$, and $\tilde{\psi}_{-}$, the eigenvector of
$(\hat{\mathfrak{h}}^{(m)}+\hat{\epsilon})\restriction\mathcal{H}_{\mathfrak{b}}^{(0+)}$
with eigenvalue $\omega_{-}$. Then the vectors $\psi_{+}$ and
$T^{-1}\tilde{\psi}_{-}=:\psi_{-}$ in $\mathcal{H}_{\mathfrak{b}}^{(0)}$ are
the eigenvectors of $\hat{\mathfrak{h}}^{(m)}$ with eigenvalues $\omega_{+}$
and $\omega_{-}$ respectively. Moreover, (89) implies

$\psi_{+}(\eta)\cong(T\psi_{-})(\eta).$ (91)

##  V The quantum dynamics

We now study the dynamics of the model for the cases of $\hat{p}_{\varphi}\neq
0$ and $\hat{p}_{\varphi}=0$ respectively. For $\hat{p}_{\varphi}\neq 0$, the
corresponding classical solution is an extension of the Schwarzschild interior
with an extra minimally coupled massless scalar field. This extension is
referred to as the Janis-Newman-Winicour (JNW) interior which differs from the
usual JNW spacetime as an extension of Schwarzschild exterior Janis et al.
(1968). In the classical JNW interior Zhang and Zhang (2020), characterized by
a parameter $B=2\sqrt{m^{2}+Gp_{\varphi}^{2}/4\pi}$, there are two
singularities at $r=0$ and $r=B$ respectively, where $r$ is the radial
coordinate. Once $p_{\varphi}$ vanishes, the singularity at $r=B$ will
disappear and is replaced by the Schwarzschild horizon, so that the JNW
interior becomes the Schwarzschild interior.

In Section II.1, gravity is deparametrized by the scalar field which provides
a material reference frame of time. However, for the case of
$\hat{p}_{\varphi}=0$, the reference frame of time will disappear. The
physical Hamiltonian becomes the Hamiltonian constraint. However, it is still
necessary to understand the dynamics of the system as the relational evolution
with respect to certain gravitational degree of freedom. The information of
the dynamics is encoded in the solutions to $\hat{\mathfrak{h}}\psi=0$. Thus
we need to solve this equation.

###  V.1 Dynamics for $\hat{p}_{\varphi}\neq 0$

By (22), the Hilbert space $L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot))$ of the model consists
of functions $\psi:m\mapsto\psi(m)\in\mathcal{H}_{\mathfrak{b}}(m)$. As shown
in Section III.2, $\mathcal{H}_{\mathfrak{b}}(m)$ can be chosen as the one
defined by (31) with $\varepsilon_{b}=0$ and some $m$-dependent $\delta_{b}$,
denoted by $\delta^{(m)}_{b}$ whose explicit expression depends on the schemes
to quantize $\mathfrak{h}$. With this convention, a state $\psi\in
L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}};\mathcal{H}_{\mathfrak{b}}(\cdot))$ can be represented by
a family of functions

$\psi(m,\cdot):\eta\to\psi(m,2\delta_{b}^{(m)}\eta)$

with $\eta\in\mathbb{Z}$. Given a state $\psi$, according to (9) and (11), an
associated dynamical state reads

$\psi(\varphi,m)=e^{\mp i\frac{\sqrt{4\pi
G}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi\sqrt{\hat{\mathfrak{h}}^{(m)}}}\hat{P}_{[0,\infty)}\psi(m).$
(92)

Let $|\omega(m)\rangle\in\mathcal{H}_{\mathfrak{b}}(m)$ be the normalized
eigenvector of $\hat{\mathfrak{h}}^{(m)}$ with respect to the eigenvalue
$\omega(m)$. Then (92) is simplified as

$\psi(\varphi,m)=\sum_{\omega(m)\geq 0}e^{\mp i\frac{\sqrt{4\pi
G\omega(m)}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi}\langle\omega(m)|\psi(m)\rangle|\omega(m)\rangle.$
(93)

We choose $\psi(m)$ as

$\psi(m,\eta)=e^{-\frac{(m-m_{0})^{2}}{2\sigma_{m}^{2}}+i\lambda
m}e^{-\frac{(\eta-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta}$ (94)

which carries some semiclassical features. According to (92), the initial
state evolved by the Hamiltonian is $\hat{P}_{[0,\infty)}\psi(m)$. Then, it is
possible that $\hat{P}_{[0,\infty)}\psi(m)$ is no longer a semiclassical wave
packet even though $\psi(m)$ is. To see how to avoid this possibility, we
introduce the expectation value of $\hat{\mathfrak{h}}^{(m)}$,

$\omega_{o}(m):=\langle\psi(m)|\hat{\mathfrak{h}}^{(m)}|\psi(m)\rangle$

and its uncertainty

$\Delta\omega(m):=\sqrt{\langle\psi(m)|(\hat{\mathfrak{h}}^{(m)})^{2}|\psi(m)\rangle-\langle\psi(m)|\hat{\mathfrak{h}}^{(m)}|\psi(m)\rangle^{2}}.$

For each $m$, we think of $\psi(m)$ as a wave function of the eigenvalues
$\omega$ of $\hat{\mathfrak{h}}^{(m)}$, so it is some wave packet peaked at
$\omega_{0}(m)$ with fluctuation $\Delta\omega(m)$. The projection
$\hat{P}_{[0,\infty)}$ cuts off $\psi(m)$ at $m=0$ and vanishes it for all
$\omega(m)\leq 0$. Therefore, $\hat{P}_{[0,\infty)}\psi(m)$ can keep the wave-
packet feature of $\psi(m)$ only if $|\omega_{o}(m)|\gg\Delta\omega(m)$. This
condition is the criterion to choose the parameters in (94).

For a properly chosen $\psi(m)$, its evolution reads

$\psi(m,\eta,\varphi)=e^{-\frac{(m-m_{0})^{2}}{2\sigma_{m}^{2}}+i\lambda
m}\sum_{\omega(m)\geq 0}\langle\mu_{\eta}^{(m)}|\omega(m)\rangle
e^{i\frac{\sqrt{4\pi
G\omega(m)}}{L_{0}\gamma\ell_{\mathrm{p}}^{2}}\varphi}\sum_{\eta^{\prime}}e^{-\frac{(\eta^{\prime}-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta^{\prime}}\langle\omega(m)|\mu_{\eta^{\prime}}^{(m)}\rangle.$
(95)

To check the consistence between the quantum dynamics and effective dynamics,
we calculate the expectation value of the operator $\hat{p}_{b}$ as

$\displaystyle\frac{\langle\hat{p}_{b}\rangle}{\gamma\delta_{b}\ell_{\mathrm{p}}^{2}}=\mathcal{N}^{2}\int_{-\infty}^{\infty}{\rm
d}me^{-\frac{(m-m_{0})^{2}}{\sigma_{m}^{2}}}\sum_{\eta}\eta\left|\sum_{\omega(m)\geq
0}\langle\mu_{\eta}^{(m)}|\omega(m)\rangle e^{i\frac{\sqrt{4\pi
G\omega(m)}}{\ell_{\mathrm{p}}^{2}\gamma
L_{0}}\varphi}\sum_{\eta^{\prime}}e^{-\frac{(\eta^{\prime}-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta^{\prime}}\langle\omega(m)|\mu_{\eta^{\prime}}^{(m)}\rangle\right|^{2}$
(96) $\displaystyle=:$
$\displaystyle\frac{1}{\sqrt{\pi}\,\sigma_{m}}\sum_{\eta}\eta\int_{-\infty}^{\infty}{\rm
d}me^{-\frac{(m-m_{0})^{2}}{\sigma_{m}^{2}}}\mathcal{P}(\eta,m,\varphi)$

where the normalization factor $\mathcal{N}$ is given by

$\mathcal{N}^{-2}=\int_{-\infty}^{\infty}{\rm
d}me^{-\frac{(m-m_{0})^{2}}{\sigma_{m}^{2}}}\sum_{\omega(m)\geq
0}\left|\sum_{\eta}e^{-\frac{(\eta-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta}\langle\omega(m)|\mu_{\eta}^{(m)}\rangle\right|^{2}=:\int_{-\infty}^{\infty}{\rm
d}me^{-\frac{(m-m_{0})^{2}}{\sigma_{m}^{2}}}\mathfrak{n}(m)$ (97)

The integral in (96) can be calculated by using the saddle point approximation
as

$\displaystyle\frac{\langle\hat{p}_{b}\rangle}{\gamma\delta_{b}\ell_{\mathrm{p}}^{2}}=$
$\displaystyle\sum_{\eta}\eta\,\mathcal{P}(\eta,m_{0},\varphi)\left(1+\frac{\sigma_{m_{0}}^{2}}{4}\frac{\mathcal{P}^{\prime\prime}(\eta,m_{0},\varphi)}{\mathcal{P}(\eta,m_{0},\varphi)}+\cdots\right).$
(98)

Therefore, as far as the leading order of the evolution is concerned, we only
need to compute $\mathcal{P}(\eta,m_{0},\varphi)$. The result of (97) reads

$\mathcal{N}^{-2}=\sqrt{\pi}\,\sigma_{m_{0}}\mathfrak{n}(m_{0})\left(1+\frac{\sigma_{m}^{2}}{4}\frac{\mathfrak{n}^{\prime\prime}(m_{0})}{\mathfrak{n}(m_{0})}+\cdots\right).$
(99)

Hence one has

$\displaystyle\mathcal{P}(\eta,m_{0},\varphi)\cong\frac{\left|\sum_{\omega(m_{0})\geq
0}\langle\mu_{\eta}^{(m_{0})}|\omega(m_{0})\rangle e^{i\frac{\sqrt{4\pi
G\omega(m_{0})}}{\ell_{\mathrm{p}}^{2}\gamma
L_{0}}\varphi}\sum_{\eta^{\prime}}e^{-\frac{(\eta^{\prime}-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta^{\prime}}\langle\omega(m_{0})|\mu_{\eta^{\prime}}^{(m_{0})}\rangle\right|^{2}}{\sum_{\omega(m_{0})\geq
0}\left|\sum_{\eta}e^{-\frac{(\eta-\eta_{0})^{2}}{2\sigma_{\eta}^{2}}-i\beta\eta}\langle\omega(m_{0})|\mu_{\eta}^{(m_{0})}\rangle\right|^{2}}.$
(100)

By this formula, $\mathcal{P}(\eta,m_{0},\varphi)$ can be computed numerically
easily. The numerical results of $\mathcal{P}(\eta,m_{0},\varphi)^{1/2}$ and
$\langle\hat{p}_{b}\rangle/(\gamma\delta_{b}\ell_{\mathrm{p}}^{2})\cong\sum_{\eta}\eta\mathcal{P}(\eta,m_{0},\varphi)$
are shown in Fig. 1, where we choose $\delta_{b}=\sqrt{\Delta}$ with $\Delta$
being the area gap in LQG. Moreover, one can also calculate the evolution of
$p_{b}$ with respect to the effective Hamiltonian

$H=\frac{4\pi}{L_{0}G\gamma}\sqrt{\frac{2}{\delta_{b}\delta_{c}}p_{b}\sin(\delta_{b}b)p_{c}\sin(\delta_{c}c)+\frac{1}{\delta_{b}^{2}}p_{b}^{2}\sin(\delta_{b}b)^{2}+\gamma^{2}p_{b}^{2}}.$
(101)

As a comparison, the results of the effective dynamics and the classical
dynamics are also plotted in Fig. 1.

As shown in Fig. 1, both of the classical singularities could be resolved by
the effective dynamics where $|\eta|$ is prevented from reaching $0$ by the
bounces at the local minimums. Then the classical spacetime is extended
periodically. The quantum evolution matches very well with the effective
dynamics for several periods around $\varphi=0$ when the semiclassical feature
of the coherent state is well kept. The effective evolution and thus the
quantum evolution match well with the classical dynamics in the classical
regime. Thus the current quantum theory has a correct semiclassical limit and
its semiclassical features can be responded properly by the effective
dynamics. However, the coherent property of the state cannot be kept along the
whole evolution, since the width of the wave packet grows as the time
$\varphi$ runs far away from the initial value $\varphi=0$. This leads to a
significant difference between the quantum and effective dynamics in late
time.

Figure 1: Plots of the evolution of the wave packet (top panel) and the
evolution of $|\eta|=p_{b}/(\gamma\delta_{b}\ell_{\mathrm{p}}^{2})$ derived by
the quantum, the effective dynamics and the classical dynamics (bottom panel).
According to the results, the quantum dynamics as well as the effective
dynamics matches well with the classical dynamics in the classical regime.
Thus the quantum model admits a correct classical limit. Moreover, the quantum
evolution of $p_{b}$ matches very well with the effective dynamics in the
domain where the wave packet is sharply peaked. However, the width of the wave
packet grows as $\varphi$ goes beyond the domain. Then the effective dynamics
is no longer valid.

###  V.2 Dynamics for $\hat{p}_{\varphi}=0$

In the case of $\hat{p}_{\varphi}=0$, the dynamics is encoded in the equation

$\hat{\mathfrak{h}}\psi=0.$ (102)

Alternatively, one could also consider the Hamiltonian constraint operator
corresponding to the vacuum Hamiltonian constraint multiplied by volume as a
lapse function Zhang et al. (2020). To solve (102), it is convenient to
represent $\psi(m)$ for each $m$ by
$\psi(m,\cdot):\omega(m)\mapsto\psi(m,\omega(m))\in\mathbb{C}$ with
$\omega(m)\in\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})$. Then the action of
$\overline{\hat{\mathfrak{h}}}$ on $\psi$ reads

$(\hat{\mathfrak{h}}\psi)(m,\omega(m))=\omega(m)\psi(m,\omega(m)).$ (103)

By (23) the inner product of two states $\psi_{1}$ and $\psi_{2}$ reads

$\left(\psi_{1},\psi_{2}\right)=\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\sum_{\omega(m)\in\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})}\psi_{1}(m,\omega(m))^{*}\psi_{2}(m,\omega(m))$
(104)

where ∗ means the complex conjugate.

Given a solution $\psi$ to (102), Eq. (103) implies

$\psi(m,\omega(m))=0,\forall\omega(m)\neq 0.$ (105)

Thus the support of the solution $\psi$ is contained in the set

$\mathcal{S}_{0}=\\{(m,\omega(m))|m\in\sigma_{\mathfrak{c}},\omega(m)\in\sigma(\hat{\mathfrak{h}}^{(m)}),\omega(m)=0\\},$
(106)

which is identified naturally to the set

$\sigma_{\xi}:=\\{m\in\sigma_{\mathfrak{c}},0\in\sigma(\hat{\mathfrak{h}}^{(m)})\\}.$
(107)

Given two functions $\psi_{i}$ with $i=1,2$ on $\sigma_{\xi}$, (104) can be
expressed by

$(\psi_{1},\psi_{2})=\int_{\sigma_{\xi}}{\rm
d}\mu_{\mathfrak{c}}\psi_{1}(m)^{*}\psi_{2}(m).$ (108)

Thus, one could naively deem that the functions on $\sigma_{\xi}$ with the
inner product (108) would constitute the physical Hilbert space. However,
depending on the explicit expression of $\mu_{\mathfrak{c}}$, it could occur
that the right hand side of (108) vanishes for regular functions $\psi_{1}$
and $\psi_{2}$. To see how this happens, let us assume
$\sigma_{\mathfrak{c}}=\mathbb{R}$ at first. Because the eigenvalues of
$\hat{\mathfrak{h}}^{(m)}$ can be expanded at some $m_{o}$ closed to $m$ by a
power series of $m-m_{o}$ and $\sigma(m_{o})$ is discrete, in general they are
not $0$. Hence it is reasonable to expect that there are only countably many
$m\in\sigma_{\mathfrak{c}}$ such that $0\in\sigma(\hat{\mathfrak{h}}^{(m)})$.
Then, both $\sigma_{\xi}$ and $\mathcal{S}_{0}$ are countable sets. This
speculation is confirmed by our numerical computation in the $\mu_{o}$-scheme
as we as the scheme with $\delta_{c}=\sqrt{\Delta}$ and
$\delta_{b}=\sqrt{\Delta}/(2|m|)$, as shown in Fig. 2. For the case of
$\sigma_{\mathfrak{c}}\neq\mathbb{R}$, one has
$\sigma_{\mathfrak{c}}\subset\mathbb{R}$ because
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ was assumed to be self-adjoint.
Then the resulting $\sigma_{\xi}$ is just a subset of that for the case of
$\sigma_{\xi}=\mathbb{R}$. Therefore, $\sigma_{\xi}$ is always countable.
Since $\sigma_{\xi}$ is countable, it could occur that
$\sigma_{\xi}\subset\sigma_{\mathfrak{c}}$ is of vanishing measure, i.e.
$\mu_{\mathfrak{c}}(\sigma_{\xi})=0$. For this case, the right hand side of
(108) will vanish and thus it cannot define an inner product for the physical
Hilbert space. Hence we introduce the following procedure to define the inner
product in the solution space, which is valid not only for the case of
$\mu_{\mathfrak{c}}(\sigma_{\xi})=0$ but also for the case of
$\mu_{\mathfrak{c}}(\sigma_{\xi})\neq 0$. Let $\tilde{\delta}(m_{o},\cdot)$
for each $m_{o}\in\sigma_{\mathfrak{c}}$ be the function on
$\sigma_{\mathfrak{c}}$ such that (i) $\tilde{\delta}(m_{o},m)=0$ for all
$m\neq m_{o}$ and (ii) $\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\tilde{\delta}(m_{o},m)=1$. Thus
$\tilde{\delta}(m_{o},\cdot)$ is the Dirac-$\delta$ distribution for
$\mu_{\mathfrak{c}}(\sigma_{\xi})=0$ and proportional to the Kronecker
$\delta$ function otherwise. Given a regular function $\psi(m,\omega(m))$, a
solution to (102) can be generated as

$\Psi(m)=\sum_{n}\tilde{\delta}(m_{o}^{(n)},m)\delta_{0,\omega_{m}}\psi(m,\omega(m)),$
(109)

where $m_{o}^{(n)}\in\sigma_{\xi}\subset\sigma_{\mathfrak{c}}$. By choosing an
appropriate dense subspace $\mathcal{S}\subset\mathcal{H}$, these $\Psi$ of
(109) are indeed anti-linear functionals on $\mathcal{S}$ as

$\Psi:\phi\mapsto\Psi[\phi]:=\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\sum_{\omega_{m}}\Psi(m)^{*}\phi(m,\omega_{m})=\sum_{m_{o}^{(n)}}\psi(m_{o}^{(n)},0)^{*}\phi(m_{o}^{(n)},0)$
(110)

for all $\phi\in\mathcal{S}$. Hence $\Psi$ is the solution to (102) in the
sense that $\Psi[\hat{\mathfrak{h}}\phi]=0$ for all $\phi\in\mathcal{S}$. Thus
(109) defines a rigging map $\mathbb{P}:\psi\mapsto\mathbb{P}\psi$ on
$\mathcal{S}$. Therefore, by the refined algebraic quantization procedure
Thiemann (2008), the physical inner product of two solutions
$\Psi_{i}=\mathbb{P}\psi_{i}$ with $i=1,2$ reads

$(\Psi_{1}|\Psi_{2})=\Psi_{1}[\psi_{2}]=\sum_{n\in\mathbb{Z}}\psi_{1}(m_{o}^{(n)})^{*}\psi_{2}(m_{o}^{(n)}),$
(111)

which coincides with (108) if $\mu_{\mathfrak{c}}(\sigma_{\xi})\neq 0$. Hence
the physical Hilbert space of the solutions is given by

$\mathcal{H}_{\rm BH}:=\overline{\\{f:\sigma_{\xi}\to\mathbb{C},\
\sum_{n}|f(m_{o}^{(n)})|^{2}<\infty\\}}.$ (112)

The Dirac observable
$\hat{\xi}_{\delta_{c}}:=\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ in
$\mathcal{H}$ can be promoted to an operator $\hat{\xi}^{\prime}_{\delta_{c}}$
in $\mathcal{H}_{\rm BH}$ by the dual action such that

$(\hat{\xi}_{\delta_{c}}^{\prime}f)(m_{o}^{(n)})=L_{0}\gamma\delta_{c}m_{o}^{(n)}f(m_{o}^{(n)}),\
\forall\ m_{o}^{(n)}.$ (113)

Eq. (113) implies that each $m_{o}^{(n)}$ is an eigenvalue of
$\xi_{\delta_{c}}^{\prime}$, and $\xi_{\delta_{c}}^{\prime}$ is self-adjoint
in $\mathcal{H}_{\rm BH}$ with the spectrum $\overline{\sigma_{\xi}}$ as the
closure of $\sigma_{\xi}$.

Given $\psi\in\mathcal{H}_{\mathfrak{b}}(m)$, Eq. (83) indicates

$(\overline{\hat{\mathfrak{h}}^{(-m)}}\psi)(\eta)^{*}=(\overline{\hat{\mathfrak{h}}^{(m)}}\psi^{*})(\eta)$
(114)

where $\psi^{*}\in\mathcal{H}_{\mathfrak{b}}(m)$ is defined by
$\psi^{*}(\eta)=\psi(\eta)^{*}$. For a given $m_{o}^{(n)}\in\sigma_{\xi}$, let
$\psi_{0}$ be the eigenvector of
$\overline{\hat{\mathfrak{h}}^{(m_{o}^{(n)})}}$ with the eigenvalue $0$. Then
(114) ensures that

$\overline{\hat{\mathfrak{h}}^{(-m_{o}^{(n)})}}\psi_{0}^{*}=\overline{\hat{\mathfrak{h}}^{(m_{o}^{(n)})}}\psi_{0}=0.$
(115)

Thus, $\psi_{0}^{*}$ is an eigenvector of
$\hat{\mathfrak{h}}^{(-m_{o}^{(n)})}$ with the eigenvalue $0$. Therefore, one
has $-m_{o}^{(n)}\in\overline{\sigma_{\xi}}$ provided
$m_{o}^{(n)}\in\overline{\sigma_{\xi}}$.

For $m=0$, one has the operator

$\hat{\mathfrak{h}}^{(0)}=\frac{1}{\delta_{b}^{2}}\hat{\beta}_{\delta_{b}}^{2}+\gamma^{2}\hat{p}_{b}^{2}.$

Assume that there is an eigenvector $\phi$ of
$\overline{\hat{\mathfrak{h}}^{(0)}}$ with the eigenvalue $0$. Because of
$\hat{\beta}_{\delta_{b}}^{2}\geq 0$ and $\hat{p}_{b}^{2}\geq 0$, $\phi$ would
satisfy

$\hat{\beta}_{\delta_{b}}^{2}\phi=0=\hat{p}_{b}^{2}\phi.$ (116)

By the definitions of $\hat{\beta}_{\delta_{b}}$ and $\hat{p}_{b}^{2}$, one
can easily check that there is no nontrivial $\phi$ satisfying (116). Hence,
$0$ is not an eigenvalue of $\overline{\hat{\mathfrak{h}}^{(0)}}$. Taking
account of Theorem IV.1, one gets the conclusion that
$0\notin\sigma(\hat{\mathfrak{h}}^{(m)})$ for sufficiently small $|m|$.
Therefore, there exists a gap between the spectrum $\overline{\sigma_{\xi}}$
and $0$, i.e. $0\notin\overline{\sigma_{\xi}}$. Note that
$\overline{\sigma_{\xi}}$ is the spectrum of
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ whose classical limit is
proportional to the ADM mass of the Schwarzschild BH. Thus, this analysis
shows the existence of discrete mass spectrum of the BH. If certain mechanism
of BH evaporation could be introduced into our quantum model consistently, the
BH would have to evaporate its masses discretely, and the evaporation would
have to eventually halt at some state corresponding to the non-vanishing
minimal mass, which is referred to as the BH remnant.

Figure 2: Plots of $\sigma_{\xi}$ for the $\mu_{o}$ scheme with
$\delta_{b}=\sqrt{\Delta}=\delta_{c}$ (top panel) and the scheme with
$\delta_{b}=\sqrt{\Delta}/(2\,|m|)$ and $\delta_{c}=\sqrt{\Delta}$ (bottom
panel). As shown in the figure, nearby each value of $m_{o}^{(n)}$ there
exists an adjoint value of $m_{o}^{(n^{\prime})}$. $0$ does not belong to
$\sigma_{\xi}$. The parameters are chosen as $\gamma=0.2374$,
$\Delta=4\sqrt{3}\pi\gamma\ell_{\mathrm{p}}^{2}$ and $\ell_{\mathrm{p}}=1$.

The above analysis is compatible with the numerical results in Fig. 2, which
shows that the values of $m_{o}^{(n)}$ are discrete and have the following
characters. First, for each $m_{o}^{(n)}$, there exists an adjoint
$m_{o}^{(n^{\prime})}$ nearby it. This property comes from the symmetric
property (91) of the eigenvectors of $\hat{\mathfrak{h}}^{(m)}$. Second, the
lowest value of $|m_{o}^{(n)}|$ in $\sigma_{\xi}$ can be obtained as
$|m_{o}^{(\rm lwt)}|=0.5499$ for the $\mu_{o}$ scheme and $|m_{o}^{(\rm
lwt)}|=0.5362$ for the other scheme, where the parameters are chosen as
$\gamma=0.2374$, $\Delta=4\sqrt{3}\pi\gamma\ell_{\mathrm{p}}^{2}$ and
$\ell_{\mathrm{p}}=1$.

##  VI Concluding Remark

The loop quantization of the model of Schwarzschild interior coupled to a
massless scalar field has been studied in the previous sections. By applying
the deparametrization procedure, we get the physical Hamiltonian
$\mathfrak{h}$ of this model with respect to the scalar field. Since $p_{c}c$
is a Dirac observable in the classical theory, $\mathfrak{h}$ is promoted to
an operator $\hat{\mathfrak{h}}$ commutating with the operator
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ which corresponds to $p_{c}c$.
Replacing $\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ in
$\hat{\mathfrak{h}}$ with its spectrum $L_{o}\gamma
m\in\sigma_{\mathfrak{c}}$, we obtain a family of operators
$\hat{\mathfrak{h}}^{(m)}$. It is shown that both $\hat{\mathfrak{h}}^{(m)}$
and $\hat{\mathfrak{h}}$ are self-adjoint. The spectrum of
$\hat{\mathfrak{h}}^{(m)}$ and its analyticity with respect to $m$ are
studied. Moreover, we develop a numerical method to diagonalize
$\hat{\mathfrak{h}}^{(m)}$. Based on these results, the dynamics for the cases
of $\hat{p}_{\varphi}\neq 0$ and $\hat{p}_{\varphi}=0$ are studied
respectively.

For the case of $\hat{p}_{\varphi}\neq 0$, the evolution of a wave packet is
considered and the results are compared with the effective dynamics governed
by the effective Hamiltonian. It turns out that the quantum evolution matches
well with the effective dynamics in the domain where the wave packet is
sharply peaked. However, the width of the wave packet would increase as the
relational time $\phi$ evolves. Thus an inconsistence between the quantum
dynamic and the effective dynamics would occur at late time. The numerical
codes to compute the evolution can be found in git .

For the case of $\hat{p}_{\varphi}=0$, the constraint $\hat{\mathfrak{h}}=0$
is imposed to get the physical states of the loop quantum Schwarzschild
interior model. Its physical Hilbert space $\mathcal{H}_{\rm BH}$ is obtained.
The spectrum $\overline{\sigma_{\xi}}$ of the Dirac observable
$\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$ in $\mathcal{H}_{\rm BH}$ is
analyzed by both analytical and numerical methods. It turns out that the
$\overline{\sigma_{\xi}}$ is discrete and it does not contain $0$, provided
that the parameters $\delta_{b}^{(m)}$ and $\varepsilon_{b}(m)$ satisfy the
suitable analyticity. Thus, there exists a gap between
$\overline{\sigma_{\xi}}$ and $0$. Moreover, by the numerical method to
diagonalize $\hat{\mathfrak{h}}$, we can also compute
$\overline{\sigma_{\xi}}$ numerically git . Some interesting results would be
obtained if certain mechanism of BH evaporation could be introduced into our
quantum model consistently. The only possibility is that the BH evaporates its
mass discretely since $\overline{\sigma_{\xi}}$ is discrete. Moreover, because
of $0\notin\overline{\sigma_{\xi}}$, the evaporation would eventually halt at
the state corresponding to the non-vanishing minimal mass. Hence, our result
supports the existence of a black hole remnant after Hawking evaporation.

It should be noted that the analyses in the current paper are valid for a
quite general class of schemes such that (i) a separable Hilbert subspace
$\mathcal{H}_{\mathfrak{c}}\subset\tilde{\mathcal{H}}_{\mathfrak{c}}$ can be
chosen to define an operator $\widehat{p_{c}\sin(\delta_{c}c)/\delta_{c}}$
corresponding to $p_{c}\sin(\delta_{c}c)/\delta_{c}$, which is self-adjoint
and commutates with the physical Hamiltonian $\hat{\mathfrak{h}}$; and (ii)
the quantum parameter $\delta_{b}$ is a constant or any function of
$p_{c}\sin(\delta_{c}c)/\delta_{c}$. With the numerical method developed in
this paper, it becomes possible to further study the Hawking radiation with
the matter backreaction and the distortion of the Hawking spectrum in details.
We leave this open issue for our future works. Thus the discrete mass spectrum
predicted by our LQG model provides a solid starting point to study the
possibilities of considering the BH remnants as dark matter candidates, as
well as solving the puzzle of information loss in BH evaporation.

##  VII acknowledgements

This work is supported by NSFC with Grants No. 11961131013, No. 11875006 and
No. 11775082. CZ acknowledges the support by the Polish Narodowe Centrum
Nauki, Grant No. 2018/30/Q/ST2/00811.

##  Appendix A Asymptotic behavior of $\hat{\beta}_{\delta_{b}}$

By definition, the eigen-equation of $\hat{\beta}_{\delta_{b}}$ implies that
its eigen-functions satisfy

$\psi(\mu+2\delta_{b})=\frac{4i\omega}{\gamma\ell_{\mathrm{p}}^{2}}\frac{1}{\mu+2\delta_{b}}\psi(\mu)+\frac{\mu}{\mu+2\delta_{b}}\psi(\mu-2\delta_{b}),$
(117)

which can be rewritten as

$\vec{\psi}(\mu+2\delta_{b})=A(\mu)\vec{\psi}(\mu)$ (118)

with

$\vec{\psi}(\mu+2\delta_{b}):=\left(\begin{array}[]{c}\psi(\mu+2\delta_{b})\\\
\psi(\mu)\end{array}\right)\quad\text{ and }\quad
A(\mu):=\left(\begin{array}[]{cc}\frac{4i\omega}{\gamma\ell_{\mathrm{p}}^{2}}\frac{1}{\mu+2\delta_{b}}&\frac{\mu}{\mu+2\delta_{b}}\\\
1&0\end{array}\right).$

Define

$B(\mu)=:\left(\begin{array}[]{cc}\psi_{0}^{+}(\mu+2\delta_{b})&\psi_{0}^{-}(\mu+2\delta_{b})\\\
\psi_{0}^{+}(\mu)&-\psi_{0}^{-}(\mu)\end{array}\right)$

where

$\psi_{0}^{\pm}(\mu)=\frac{1}{\sqrt{|\mu|}}e^{\pm ik\ln(|\mu|)}.$

We obtain that the vector-valued function
$\vec{\chi}(\mu)=\left(\begin{array}[]{c}\chi_{1}(\mu)\\\
\chi_{2}(\mu)\end{array}\right)$, defined by

$\vec{\psi}(\mu+2\delta_{b})=:B(\mu)\vec{\chi}(\mu+2\delta_{b}),$ (119)

satisfies

$\vec{\chi}(\mu+2\delta_{b})=B(\mu)^{-1}A(\mu)B(\mu-2\delta_{b})\vec{\chi}(\mu):=M(\mu)\vec{\chi}(\mu).$
(120)

which is obtained by applying (118). Our purpose is to derive the condition
with which the vector-valued function $\vec{\chi}(\mu)$ in right-hand-side of
(119) can be approximated by some constant up to some $O(\mu^{-1})$ term. This
can be achieved if the matrix $M(\mu)$ satisfies

$M(\mu)=M+O(|\mu|^{-2}),$ (121)

with some constant $M$. Substituting (121) into (120), we obtain

$k=\frac{\omega}{\gamma\ell_{\mathrm{p}}^{2}\delta_{b}}$ (122)

with

$M(\mu)=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)+O(|\mu|^{-2}).$
(123)

Therefore, one gets

$\psi(\mu)=\left\\{\begin{aligned}
&\chi_{1}\psi_{0}^{+}(\mu)+\chi_{2}\psi_{0}^{-}(\mu),\
\mu=\varepsilon_{b}+4n\delta_{n}\\\
&\chi_{1}\psi_{0}^{+}(\mu)-\chi_{2}\psi_{0}^{-}(\mu),\
\mu=\varepsilon_{b}+4n\delta_{n}+2\delta_{b}\end{aligned}\right.$ (124)

with $n\in\mathbb{Z}$.

##  Appendix B The self-adjointness of $\hat{\mathfrak{h}}^{(m)}$ and
$\hat{\mathfrak{h}}$

We now prove that $\hat{\mathfrak{h}}^{(m)}$ and $\hat{\mathfrak{h}}$ are
essentially self-adjoint .

Define an operator $\hat{A}$ as

$\hat{A}|\mu\rangle=\left(\frac{1}{\delta_{b}^{2}}\frac{\gamma^{2}\ell_{\mathrm{p}}^{4}}{16}\big{(}(\mu+2\delta_{b})^{2}+\mu^{2}\big{)}+\frac{1}{4}\gamma^{4}\ell_{\mathrm{p}}^{4}(\mu-2\mathfrak{b})^{2}\right)|\mu\rangle=:A(\mu)|\mu\rangle,$
(125)

whose domain reads

$D(\hat{A})=\left\\{|\psi\rangle,\,\sum_{\mu}\left|A(\mu)\psi(\mu)\right|^{2}<\infty\right\\}.$
(126)

Then $\hat{A}$ is self-adjoint because $A(\mu)$ is real. Let $\hat{B}$ be the
operator defined on the domain $D_{\mathfrak{b}}$ as

$\hat{B}=\frac{2mL_{0}\gamma}{\delta_{b}}\hat{\beta}_{\delta_{b}}+\frac{1}{\delta_{b}^{2}}\left(\hat{\beta}_{\delta_{b}}^{2}-\frac{1}{4}\left(\hat{p}_{b}^{2}+(\hat{p}_{b}+\delta_{b}\gamma\ell_{\mathrm{p}}^{2})^{2}\right)\right)$
(127)

Then $\hat{B}$ can be expressed as

$\hat{B}=\frac{2mL_{0}\gamma}{\delta_{b}}\frac{1}{2i}(\alpha^{\dagger}-\alpha)-\frac{1}{4\delta_{b}^{2}}((\alpha^{\dagger})^{2}+\alpha^{2})$
(128)

where

$\displaystyle\alpha^{\dagger}\,|\mu\rangle$
$\displaystyle=\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}(\mu+2\delta_{b})\,|\mu+2\delta_{b}\rangle$
(129) $\displaystyle\alpha\,|\mu\rangle$
$\displaystyle=\frac{\gamma\ell_{\mathrm{p}}^{2}}{2}\mu\,|\mu+2\delta_{b}\rangle.$

Given $\psi\in D_{\mathfrak{b}}$, we have

$\|\hat{B}\psi\|\leq\left\|\left(\frac{2mL_{0}\gamma}{\delta_{b}}\frac{1}{2i}\alpha^{\dagger}-\frac{1}{4\delta_{b}^{2}}(\alpha^{\dagger})^{2}\right)\psi\right\|+\left\|\left(\frac{2mL_{0}\gamma}{\delta_{b}}\frac{1}{2i}\alpha+\frac{1}{4\delta_{b}^{2}}\alpha^{2}\right)\psi\right\|.$
(130)

Moreover, a straight-forward calculation gives

$\displaystyle\left\|\left(\frac{2mL_{0}\gamma}{\delta_{b}}\frac{1}{2i}\alpha^{\dagger}-\frac{1}{4\delta_{b}^{2}}(\alpha^{\dagger})^{2}\right)\psi\right\|^{2}\leq\sum_{\mu}\Big{(}\frac{4m^{2}L_{0}^{2}\gamma^{4}\ell_{\mathrm{p}}^{4}}{16\delta_{b}^{2}}\mu^{2}+\frac{\gamma^{4}\ell_{\mathrm{p}}^{8}}{256\delta_{b}^{4}}\mu^{2}(\mu-2\delta_{b})^{2}|$
(131)
$\displaystyle\quad\quad+\frac{\gamma^{2}\ell_{\mathrm{p}}^{4}}{16\delta_{b}^{2}}\frac{2mL_{0}\gamma}{\delta_{b}}\frac{\gamma\ell_{\mathrm{p}}^{2}}{4}\left(\mu^{2}(\mu+2\delta_{b})+(\mu-2\delta_{b})^{2}\mu\right)\Big{)}|\psi(\mu)|^{2}$
$\displaystyle=:$ $\displaystyle\sum_{\mu}B^{+}(\mu)|\psi(\mu)|^{2},$

and

$\displaystyle\left\|\left(\frac{2mL_{0}\gamma}{\delta_{b}}\frac{1}{2i}\alpha-\frac{1}{4\delta_{b}^{2}}\alpha^{2}\right)\psi\right\|^{2}\leq\sum_{\mu}\Big{(}\frac{4m^{2}L_{0}^{2}\gamma^{4}\ell_{\mathrm{p}}^{4}}{16\delta_{b}^{2}}(\mu+2\delta_{b})^{2}+\frac{\gamma^{4}\ell_{\mathrm{p}}^{8}}{256\delta_{b}^{4}}(\mu+4\delta_{b})^{2}(\mu+2\delta_{b})^{2}$
(132)
$\displaystyle\quad\quad+\frac{1}{64\delta_{b}^{2}}\frac{2mL_{0}\gamma^{4}\ell_{\mathrm{p}}^{6}}{\delta_{b}}\left((\mu+2\delta_{b})^{2}\mu+(\mu+4\delta_{b})^{2}(\mu+2\delta_{b})\right)\Big{)}|\psi(\mu)|^{2}$
$\displaystyle=:$ $\displaystyle\sum_{\mu}B^{-}(\mu)|\psi(\mu)|^{2}.$

By Eqs. (130), (131) and (132) we obtain

$\displaystyle\|\hat{B}\psi\|\leq\sqrt{\sum_{\mu}B^{+}(\mu)|\psi(\mu)|^{2}}+\sqrt{\sum_{\mu}B^{-}(\mu)|\psi(\mu)|^{2}}\leq\sqrt{\sum_{\mu}4B^{-}(\mu)|\psi(\mu)|^{2}}$
(133)

Moreover, one has

$\displaystyle\|\hat{A}\psi\|=\sqrt{\sum_{\mu}A(\mu)^{2}|\psi(\mu)|^{2}}.$
(134)

By the expressions of $B^{-}(\mu)$ and $A(\mu)$, there exists some real number
$b\geq 0$ such that

$4B^{-}(\mu)\leq\frac{1}{1+2\gamma^{4}\delta_{b}^{4}}A(\mu)^{4}+b$ (135)

which implies

$\|\hat{B}\psi\|^{2}\leq\frac{1}{1+2\gamma^{4}\delta_{b}^{4}}\|\hat{A}\psi\|^{2}+b\|\psi\|^{2}$
(136)

Therefore, $\hat{\mathfrak{h}}^{(m)}=\hat{B}+\hat{A}$ is self-adjoint on
$D(A)$ and thus essentially self-adjoint on $D_{\mathfrak{b}}$ according to
the Kato-Rellich Theorem (see Theorem X.12 in Reed et al. (2003a)). Thanks to
the self-adjointness of $\hat{\mathfrak{h}}^{(m)}$, the self-adjointness of
$\hat{\mathfrak{h}}$ can be proven as follows. Let $g^{\pm}$ be elements in
the orthogonal complement of the range of $\hat{\mathfrak{h}}\pm i$
respectively, i.e., $g^{\pm}\in{\rm Ran}(\hat{\mathfrak{h}}\pm i)^{\perp}$.
Then we have

$0=\langle(\hat{A}\pm i)\psi,g^{\pm}\rangle=\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\left\langle(\hat{\mathfrak{h}}^{(m)}\pm
i)\psi(m),g^{\pm}(m)\right\rangle,\ \forall\psi\in D(\hat{\mathfrak{h}}).$
(137)

Let $f$ be an square-integrable function $f$ on $\sigma_{\mathfrak{c}}$ with
respect to $\mu_{\mathfrak{c}}$, i.e., $f\in L^{2}(\sigma_{\mathfrak{c}},{\rm
d}\mu_{\mathfrak{c}})$. Given $\psi\in D(\hat{\mathfrak{h}})$, one has

$\left|\int_{\sigma_{\mathfrak{c}}}{\rm d}\mu_{\mathfrak{c}}\left\langle
f(m)\hat{\mathfrak{h}}^{(m)}\psi(m),f(m)\hat{\mathfrak{h}}^{(m)}\psi(m)\right\rangle\right|^{2}\leq\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{c}|f(m)|^{2}\int_{\sigma_{\mathfrak{c}}}\|\hat{\mathfrak{h}}^{(m)}\psi(m)\|^{2}<\infty$
(138)

which implies that $f\psi:m\mapsto f(m)\psi(m)$ is also in
$D(\hat{\mathfrak{h}})$. Thus substituting $f\psi$ into (137), we conclude

$\int_{\sigma_{\mathfrak{c}}}{\rm
d}\mu_{\mathfrak{c}}\overline{f(m)}\left\langle(\hat{\mathfrak{h}}^{(m)}\pm
i)\psi(m),g^{\pm}(m)\right\rangle=0,\ \forall f\in
L^{2}(\sigma_{\mathfrak{c}},{\rm d}\mu_{\mathfrak{c}}).$ (139)

Therefore, it holds almost everywhere for $m$ that

$\langle(\hat{\mathfrak{h}}^{(m)}\pm i)\psi(m),g^{\pm}(m)\rangle=0.$ (140)

Since this conclusion is true for all $\psi(m)\in D_{\mathfrak{b}}(m)$, one
has that $g^{\pm}(m)\in{\rm Ran}(\hat{\mathfrak{h}}^{(m)}\pm i)^{\perp}$
almost everywhere for $m$. However, $\hat{\mathfrak{h}}^{(m)}$ is essentially
self-adjoint for all $m$. Hence one gets

${\rm Ran}(\hat{\mathfrak{h}}^{(m)}\pm i)^{\perp}=\\{0\\},\forall m.$ (141)

This ensures that $g^{\pm}(m)=0$ for all $m$. The self-adjointness of
$\hat{A}$ is thus obtained because of the basic criterion for self-adjointness
(see, e.g., Theorem VIII.3 in Reed (2003)).

##  Appendix C The analiticity of $\overline{T_{m}}$

######  Theorem C.1.

$\overline{T_{m}}$ is an analytic family of forms of type (a) in the sense of
Kato.

By definition, this theorem can be obtained directly from the following
lemmas.

######  Lemma C.1.

The norm $\|\cdot\|_{m}$ defined on $D_{\mathfrak{b}}(m_{o})$ for each $m$ is
equivalent to the norm

$\|\psi\|_{+}=\sqrt{\langle\psi|\hat{p}_{b}^{2}|\psi\rangle+\langle\psi|\psi\rangle},$
(142)

i.e., there exist constants $c,C>0$ such that

$c\|\psi\|_{+}\leq\|\psi\|_{m}\leq C\|\psi\|_{+},\forall\psi\in
D_{\mathfrak{b}}(m_{o}).$ (143)

###### Proof.

Given $|\psi\rangle=\sum_{\mu}\psi_{\mu}|\mu\rangle\in
D_{\mathfrak{b}}(m_{o})$, by (52) one gets

$\displaystyle
T_{m}(\psi,\psi)+(L_{0}^{2}m^{2}\gamma^{2}+1)\langle\psi|\psi\rangle$
$\displaystyle\geq\gamma^{2}\langle\psi|(\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}|\psi\rangle+\langle\psi|\psi\rangle\geq\gamma^{2}c\langle\psi|\hat{p}_{b}^{2}|\psi\rangle+\langle\psi|\psi\rangle$
(144)

where

$c=1+\frac{(\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}}{2}-\frac{\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b}\sqrt{4+(\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b})^{2}}}{2}>0.$
(145)

Moreover, consider the operator
$(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,(\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}+L_{0}^{2}\gamma^{2}m^{2}+1)(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}$
on $D_{\mathfrak{b}}(m_{o})$. It can be verified straight-forwardly that

$\left|\langle\mu|(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,(\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}+L_{0}^{2}\gamma^{2}m^{2}+1)(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,|\mu^{\prime}\rangle\right|<\infty.$
(146)

Thus the operator
$(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,(\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}+L_{0}^{2}\gamma^{2}m^{2}+1)(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}$
is bounded, i.e., there exist $C>0$ such that

$\left|\langle\psi|(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,(\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}+L_{0}^{2}\gamma^{2}m^{2}+1)(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}\,|\psi\rangle\right|\leq
C\langle\psi|\psi\rangle,\forall\psi\in D_{\mathfrak{b}}(m_{o}).$ (147)

By definition, $(1+\hat{p}_{b}^{2})^{-\frac{1}{2}}:D_{\mathfrak{b}}(m_{o})\to
D_{\mathfrak{b}}(m_{o})$ is surjective. Hence (147) implies

$\left|\langle\psi|(\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}+L_{0}^{2}\gamma^{2}m^{2}+1)|\psi\rangle\right|\leq
C\langle\psi|(1+\hat{p}_{b}^{2})|\psi\rangle,\forall\psi\in
D_{\mathfrak{b}}(m_{o}).$ (148)

Thus

$T_{m}(\psi,\psi)+(L_{0}^{2}m^{2}\gamma^{2}+1)\langle\psi|\psi\rangle\leq
C\langle\psi|(1+\hat{p}_{b}^{2})|\psi\rangle,\forall\psi\in
D_{\mathfrak{b}}(m_{o}).$ (149)

Then (143) is proven by (144) and (149). ∎

By Lemma C.1, for each $m$, $Q(m)$ is indeed the closure of
$D_{\mathfrak{b}}(m_{o})$ with respect to the norm $\|\cdot\|_{+}$ given in
(142). Thus one gets $Q(m)=Q(m_{o})$ for all $m$.

######  Lemma C.2.

Suppose $\delta_{b}^{(m)}$ and $\varepsilon_{b}(m)$ to be analytic functions
of $m$ at $m_{o}$ and $\delta_{b}^{(m_{o})}\neq 0$. Then
$\overline{T_{m}}(\psi,\psi)$, for each $\psi\in Q(m_{o})$, is an analytic
function of $m$ at $m_{o}$.

###### Proof.

The action of $\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}$
on $|\mu_{n}\rangle$ with $\mu_{n}=\varepsilon_{b}^{o}+2n\delta_{b}^{o}$ reads

$\displaystyle\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}|\mu_{n}\rangle=$
$\displaystyle-\frac{1}{4(\delta_{b}^{o})^{2}}(\mathfrak{f}(m)+\hat{p}_{b})(\mathfrak{f}(m)+\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\delta_{b}^{o})|\mu_{n+2}\rangle+\frac{2L_{0}m\gamma}{2\delta_{b}^{o}i}(\mathfrak{f}(m)+\hat{p}_{b})|\mu_{n+1}\rangle$
(150)
$\displaystyle+\left(\frac{1}{4(\delta_{b}^{o})^{2})}((\mathfrak{f}(m)+\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\delta_{b}^{o})^{2}+((\mathfrak{f}(m)+\hat{p}_{b})^{2})+\gamma^{2}(\delta_{b}^{(m)})^{2}((\mathfrak{f}(m)+\hat{p}_{b}-\gamma\ell_{\mathrm{p}}^{2}\mathfrak{b}/\delta_{b}^{(m)})^{2}\right)|\mu_{n}\rangle$
$\displaystyle-\frac{2L_{0}m\gamma}{2\delta_{b}^{o}i}(\mathfrak{f}(m)+\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\delta_{b}^{o})|\mu_{n-1}\rangle-\frac{1}{4(\delta_{b}^{o})^{2}}(\mathfrak{f}(m)+\hat{p}_{b}+2\gamma\ell_{\mathrm{p}}^{2}\delta_{b}^{o})(\mathfrak{f}(m)+\hat{p}_{b}+\gamma\ell_{\mathrm{p}}^{2}\delta_{b}^{o})|\mu_{n-2}\rangle$

where
$\mathfrak{f}(m):=\varepsilon_{b}(m)\delta_{b}^{o}/\delta_{b}^{(m)}-\varepsilon_{b}^{o}$
is analytic at $m_{o}$. Given $|\psi\rangle\in Q(m_{o})$, Eq. (150) implies
that
$\langle\psi|\mathfrak{i}_{m}\hat{\mathfrak{h}}^{(m)}\mathfrak{i}_{m}^{-1}|\psi\rangle$
is a finite linear combination of $\langle\psi|\hat{p}_{b}^{2}|\psi\rangle$,
$\langle\psi|\hat{p}_{b}|\psi\rangle$ and $\langle\psi|\psi\rangle$ with
coefficients depending on $m$ analytically. Since $Q(m)$ is the closure of
$D_{\mathfrak{b}}(m_{o})$ with respect to the norm $\|\cdot\|_{+}$, both
$\langle\psi|\hat{p}_{b}^{2}|\psi\rangle$ and
$\langle\psi|\hat{p}_{b}|\psi\rangle$ are well-defined for all $\psi\in Q(m)$.
Then the analyticity of $\overline{T_{m}}(\psi,\psi)$ is proven. ∎

##  Appendix D Proof of equations (62) and (63)

We notice that

$\displaystyle\lambda_{i}^{(k)}=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}\psi\rangle$
(151) $\displaystyle=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle$

because of
$\hat{P}^{(k)}\overline{\hat{\mathfrak{h}}^{(m)}}\hat{P}^{(k)}=\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)}$,
and $\hat{P}^{(k)}\psi=\psi$ for all
$\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}$.

Given $k^{\prime}>k$, one has
$\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}\subset\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}$.
Let $\tilde{\phi}_{1},\ \tilde{\phi}_{2},\
\cdots\tilde{\phi}_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}$
be some vectors such that
$\tilde{\phi}_{i-n},\tilde{\phi}_{i-n+1},\cdots,\tilde{\phi}_{i-1}\notin\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}$.
Then we have

$\displaystyle\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\tilde{\phi}_{l}\rangle=0,\forall
l=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle=\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\tilde{\phi}_{l}\rangle=0,\forall
l=1,2,\cdots,i-n-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle\leq\lambda_{i-n}^{(k)}\leq\lambda_{i}^{(k)}.$
(152)

Therefore, one obtains

$\displaystyle\lambda_{i}^{(k)}=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle$
(153) $\displaystyle=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle.$

Furthermore, for given
$\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}$,
we have

$\displaystyle\\{\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}\
\big{|}\ \|\psi\|=1;\langle\psi,\phi_{n}\rangle=0,\forall n=1,2,\cdots,i-1\\}$
(154) $\displaystyle\subset$
$\displaystyle\\{\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}\
\big{|}\ \|\psi\|=1;\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\\}.$

Thus, one has

$\displaystyle\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle\geq\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle,$
(155)

which implies

$\displaystyle\lambda_{i}^{(k)}=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle$
(156) $\displaystyle\geq$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k^{\prime})};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle=\lambda_{i}^{(k^{\prime})}.$

Similarly, we have

$\displaystyle\lambda_{i}^{(k)}=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle$
(157) $\displaystyle=$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}}\inf_{\begin{subarray}{c}\psi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle$
$\displaystyle\geq$
$\displaystyle\sup_{\phi_{1},\phi_{2},\cdots,\phi_{i-1}\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b})}}\inf_{\begin{subarray}{c}\psi\in\overline{D_{\mathfrak{b}}(m)};\|\psi\|=1\\\
\langle\psi,\phi_{n}\rangle=0,\forall
n=1,2,\cdots,i-1\end{subarray}}\langle\psi,\hat{\mathfrak{h}}^{(m)}\psi\rangle=\omega_{i}.$

Thus the proof of (62) is completed, and the existence of the limit
$\lim_{k\to\infty}\lambda_{i}^{(k)}$ can be obtained directly from (62).

##  Appendix E The finite cut-off approximation

######  Theorem E.1.

Each $\lambda_{i}$ given in (63) is an eigenvalue of
$\overline{\hat{\mathfrak{h}}^{(m)}}$. The space $\Lambda_{i}$ is an
eigenspace corresponding to the eigenvalue $\lambda_{i}$.

###### Proof.

By definition, one has

$\|\hat{P}^{(k)}(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})\psi_{i}^{(k)}\|=\|(\lambda_{i}^{(k)}-\lambda_{i})\psi_{i}^{(k)}\|\leq|\lambda_{i}^{(k)}-\lambda_{i}|,$
(158)

which implies

$\lim_{k\to\infty}\|\hat{P}^{(k)}(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})\psi_{i}^{(k)}\|=0.$
(159)

Given an arbitrary $\varphi\in D_{\mathfrak{b}}(m)$, by the definition (49) of
$D_{\mathfrak{b}}(m)$, there exists an integer $N_{\varphi}$ such that
$\varphi\in\mathcal{H}_{\mathfrak{b}}^{(\varepsilon_{b},k)},\ \forall k\geq
N_{\varphi}$. Thus, for each $\varphi\in D_{\mathfrak{b}}(m)$, one gets

$\langle\varphi|\hat{P}^{(k)}(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})|\psi_{i}^{(k)}\rangle=\langle\varphi|(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})|\psi_{i}^{(k)}\rangle,\forall
k\geq N_{\varphi}$ (160)

where we used $\hat{P}^{(k)}\varphi=\varphi$ for all $k\geq N_{\varphi}$ and
that $\hat{\mathfrak{h}}^{(m)}$ is symmetric. Hence, we have

$0=\lim_{k\to\infty}\langle\varphi|\hat{P}^{(k)}(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})|\psi_{i}^{(k)}\rangle=\langle\varphi|(\hat{\mathfrak{h}}^{(m)}-\lambda_{i})|\psi_{i}\rangle,\
\forall\varphi\in D_{\mathfrak{b}}(m).$ (161)

By this equation, $\psi_{i}$ is in the domain of the adjoint of
$\hat{\mathfrak{h}}^{(m)}$. Since $\hat{\mathfrak{h}}^{(m)}$ is essentially
self-adjoint, its adjoint is equal to its closure
$\overline{\hat{\mathfrak{h}}^{(m)}}$. Thus $\psi_{i}$ is in
$\overline{D_{\mathfrak{b}}(m)}$. Moreover, (161) also implies

$(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\psi_{i}=0,$ (162)

which ensures either $\psi_{i}=0$ or that $\psi_{i}$ is an eigenvector of
$\overline{\hat{\mathfrak{h}}^{(m)}}$ with the eigenvalue $\lambda_{i}$. We
now show that $\psi_{i}\neq 0$ and hence $\psi_{i}$ can only be an eigenvector
of $\overline{\hat{\mathfrak{h}}^{(m)}}$. Given an eigenvalue $\omega$ of
$\overline{\hat{\mathfrak{h}}^{(m)}}$, let $|\omega,\alpha\rangle$ be an
orthonormal basis of the eigenspace with respect to $\omega$. Define a
projection $\hat{P}$ as

$\hat{P}|\psi\rangle:=\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|\psi\rangle.$
(163)

Because $\overline{\hat{\mathfrak{h}}^{(m)}}$ is bounded from below and each
eigenvalue $\omega$ is of finite multiplicity, the summation in the RHS
consists of only finite terms. Then for each vectors $\psi_{i}^{(n_{l})}$ in
(64), we have

$\displaystyle\left\|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}-\sum_{\omega\in(-\infty,\lambda_{k}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|\psi_{i}\rangle(\omega-\lambda_{i})\right\|$
(164) $\displaystyle\leq$
$\displaystyle\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}\left|\left(\langle\omega,\alpha|\psi_{i}^{(n_{l})}\rangle-\langle\omega,\alpha|\psi_{i}\rangle\right)\right|\left|\omega-\lambda_{i}\right|.$

Taking account of (164), (64) and the fact that the summation contains
finitely many terms, we obtain

$\lim_{l\to\infty}\left\|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}-\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|\psi_{i}\rangle(\omega-\lambda_{i})\right\|=0,$
(165)

i.e.

$\lim_{l\to\infty}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}=\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|\psi_{i}\rangle(\omega-\lambda_{i}).$
(166)

Moreover, because of (162) and

$\displaystyle\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|\psi_{i}\rangle(\omega-\lambda_{i})=\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}|\omega,\alpha\rangle\langle\omega,\alpha|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\psi_{i}\rangle,$
(167)

we finally have

$\lim_{l\to\infty}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}=0.$
(168)

Furthermore, by using

$\|\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\psi_{i}^{(n_{l})}\|=\|(\lambda_{i}^{(n_{l})}-\lambda_{i})\psi_{i}^{(n_{l})}\|\leq|\lambda_{i}^{(n_{l})}-\lambda_{i}|,$
(169)

one has

$\lim_{l\to\infty}\|\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\psi_{i}^{(n_{l})}\|=0.$
(170)

Combining (170) with the inequality

$\displaystyle\||\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})(1-\hat{P})\psi_{i}^{(n_{l})}\|\leq\|\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\psi_{i}^{(n_{l})}\|+\|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}\|,$
(171)

we finally obtain

$\lim_{l\to\infty}\|\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})(1-\hat{P})\psi_{i}^{(n_{l})}\|\leq\lim_{l\to\infty}\|(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})\hat{P}\psi_{i}^{(n_{l})}\|=0,$
(172)

which implies

$\lim_{l\to\infty}\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})(1-\hat{P})\psi_{i}^{(n_{l})}=0.$
(173)

Defining
$\tilde{\omega}:=\inf\\{\omega\in\sigma(\hat{\mathfrak{h}}^{(m)}),\omega>\lambda_{i}\\}$,
we have

$\displaystyle\langle\hat{P}^{(n_{l})}(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})(1-\hat{P})\psi_{i}^{(n_{l})}|\psi_{i}^{(n_{l})}\rangle=\langle(1-\hat{P})(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{i})(1-\hat{P})\psi_{i}^{(n_{l})}|\psi_{i}^{(n_{l})}\rangle$
(174) $\displaystyle\geq$
$\displaystyle(\tilde{\omega}-\lambda_{i})\|(1-\hat{P})\psi_{i}^{(n_{l})}\|,$

where the last inequality is resulted from
$\langle(\overline{\hat{\mathfrak{h}}^{(m)}}-\lambda_{k})(1-\hat{P})\varphi,\varphi\rangle\geq(\tilde{\omega}-\lambda_{i})\langle(1-\hat{P})\varphi,\varphi\rangle$
for all $\varphi\in\overline{D_{\mathfrak{b}}(m)}$. The combination of (174)
and (173) leads to

$\lim_{l\to\infty}\|(1-\hat{P})\psi_{i}^{(n_{l})}\|=0.$ (175)

Furthermore, because of

$\|\psi_{i}^{(n_{l})}-\psi_{i}\|\leq\|\hat{P}\psi_{i}^{(n_{l})}-\hat{P}\psi_{i}\|+\|(1-\hat{P})\psi_{i}^{(n_{l})}\|+\|(1-\hat{P})\psi_{i}\|$
(176)

we obtain

$\lim_{l\to\infty}\|\psi_{i}^{(n_{l})}-\psi_{i}\|\leq\lim_{l\to\infty}\|\hat{P}\psi_{i}^{(n_{l})}-\hat{P}\psi_{i}\|+\|(1-\hat{P})\psi_{i}\|.$
(177)

The first term in the RHS of (177) satisfies

$\displaystyle\left\|\hat{P}\psi_{i}^{(n_{l})}-\hat{P}\psi_{i}\right\|\leq\sum_{\omega\in(-\infty,\lambda_{i}]}\sum_{\alpha}\left|\left(\langle\omega,\alpha|\psi_{k}^{(n_{l})}\rangle-\langle\omega,\alpha|\psi_{k}\rangle\right)\right|.$
(178)

Taking account of (64) and the fact that the summation in the RHS contains
finite terms, Eq. (178) implies

$\lim_{l\to\infty}\left\|\hat{P}\psi_{i}^{(n_{l})}-\hat{P}\psi_{i}\right\|=0,$
(179)

Therefore, we have

$\lim_{l\to\infty}\|\psi_{i}^{(n_{l})}-\psi_{i}\|\leq\|(1-\hat{P})\psi_{i}\|\leq\|\psi_{i}\|.$
(180)

which implies that $|\psi_{i}\rangle\neq 0$. Otherwise, one would get

$0=\lim_{l\to\infty}\|\psi_{i}^{(n_{l})}-\psi_{i}\|=\lim_{l\to\infty}\|\psi_{i}^{(n_{l})}\|,$
(181)

which is contradictory to $\|\psi_{i}^{(n)}\|=1$. Therefore, according to
(162), $\lambda_{i}$ is an eigenvalue of $\overline{\hat{\mathfrak{h}}^{(m)}}$
and $|\psi_{i}\rangle$ is a corresponding eigenvector. ∎

The above proof is inspired by the works Levitin and Shargorodsky (2004);
Lewin and Séré (2009) and (Reed, 2003, Section VIII.7).

######  Theorem E.2.

Given $\lambda_{i}$ and $\lambda_{i+1}$ as defined in (63), if
$\lambda_{i}\neq\lambda_{i+1}$, i.e. $\lambda_{i}<\lambda_{i+1}$, one has

$\sigma(\overline{\hat{\mathfrak{h}}^{(m)}})\cap(\lambda_{i},\lambda_{i+1})=\emptyset.$
(182)

###### Proof.

Consider an interval $(a,b)$ with $\lambda_{i}<a<b<\lambda_{i+1}$. By
definition of $\lambda_{i}$ and (62), there exists an integer $\tilde{N}$ such
that

$\lambda_{i}^{(k)}\leq a,\ \forall k\geq\tilde{N}.$ (183)

Then for an eigenvalue $\lambda_{i^{\prime}}^{(k)}$ of
$\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)}$ with $k\geq\tilde{N}$,

  * (i)

if $i^{\prime}\leq i$, one has

$\lambda_{i^{\prime}}^{(k)}\leq\lambda_{i}^{(k)}\leq a.$ (184)

  * (ii)

if $i^{\prime}>i$, or equivalently $i^{\prime}\geq i+1$, one has

$\lambda_{i^{\prime}}^{(k)}\geq\lambda_{i+1}^{(k)}\geq\lambda_{i}\geq b.$
(185)

The above analysis indicates

$\sigma(\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)})\cap(a,b)=\emptyset,\
\forall k\geq\tilde{N}$ (186)

where $\sigma(\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)})$ denote, as
usual, the set of eigenvalues of
$\hat{P}^{(k)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(k)}$. Let $z$ be the complex
number

$z=\frac{a+b}{2}+i\frac{a-b}{2}.$ (187)

Given $|\varphi\rangle\in D_{\mathfrak{b}}(m)$, one has
$|\psi\rangle:=(\hat{\mathfrak{h}}^{(m)}-z)|\varphi\rangle\in
D_{\mathfrak{b}}(m)$ by the expression of $\hat{\mathfrak{h}}^{(m)}$. Hence
there exists a large integer $n\geq\tilde{N}$ such that

$\hat{P}^{(n)}|\varphi\rangle=|\varphi\rangle,\
\hat{P}^{(n)}|\psi\rangle=|\psi\rangle,\
\hat{P}^{(n)}\hat{\mathfrak{h}}^{(m)}|\varphi\rangle=\hat{\mathfrak{h}}^{(m)}|\varphi\rangle.$
(188)

Then one has

$|\psi\rangle=(\hat{\mathfrak{h}}^{(m)}-z)|\varphi\rangle=(\hat{P}^{(n)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(n)}-z)|\varphi\rangle,$
(189)

which leads to

$\displaystyle(\hat{\mathfrak{h}}^{(m)}-z)^{-1}|\psi\rangle=(\hat{P}^{(n)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(n)}-z)^{-1}|\psi\rangle.$
(190)

Because of $\hat{P}^{(n)}\psi=\psi$, i.e.
$\psi\in\mathcal{H}_{\mathfrak{b}}^{(n)}$, we have

$\|(\hat{P}^{(n)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(n)}-z)^{-1}\psi\|^{2}=\sum_{i^{\prime}=1}^{2n+1}|(\lambda_{i^{\prime}}^{(n)}-z)^{-1}|^{2}|\langle\psi_{i^{\prime}}^{(n)}|\psi\rangle|^{2}$
(191)

where $\psi_{i^{\prime}}^{(n)}$ is the normalized eigenvector of
$\hat{P}^{(n)}\hat{\mathfrak{h}}^{(m)}\hat{P}^{(n)}$ corresponding to the
eigenvalue $\lambda_{i^{\prime}}^{(n)}$. According to the inequality
$\lambda_{1}^{(n)}\leq\lambda_{2}^{(n)}\leq\cdots\leq\lambda_{k}^{(n)}\leq
a\leq b\leq\lambda_{k+1}^{(n)}\leq\cdots\leq\lambda_{n}^{(n)}$, we have that

$|(\lambda_{i^{\prime}}^{(n)}-z)^{-1}|^{2}\leq\frac{2}{(a-b)^{2}}$ (192)

which implies

$\|(\hat{\mathfrak{h}}^{(m)}-z)^{-1}\psi\|^{2}\leq\frac{2}{(a-b)^{2}}\sum_{i^{\prime}=1}^{2n+1}|\langle\psi_{i^{\prime}}^{(n)}|\psi\rangle|^{2}=\frac{2}{(a-b)^{2}}\|\psi\|^{2},\
\forall\,|\psi\rangle\in D_{\mathfrak{b}}(m).$ (193)

Thus, one has

$\|(\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1}\psi\|^{2}\leq\frac{2}{(a-b)^{2}}\|\psi\|^{2},\
\forall|\psi\rangle\in\overline{D_{\mathfrak{b}}(m)}.$ (194)

As a consequence,

$\rho((\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1})\leq\frac{\sqrt{2}}{b-a}$
(195)

where $\rho((\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1})$ is the spectral
radius of $(\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1}$. Because of the self-
adjointness of $\overline{\hat{\mathfrak{h}}^{(m)}}$, the spectrum of
$(\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1}$ is

$\sigma((\overline{\hat{\mathfrak{h}}^{(m)}}-z)^{-1})=\\{(\lambda-z)^{-1},\lambda\in\sigma(\hat{\mathfrak{h}}^{(m)})\\}.$
(196)

Therefore, (195) leads to

$(a,b)\cap\sigma(\hat{\mathfrak{h}}^{(m)})=\emptyset.$ (197)

which is true for any $(a,b)\subset(\lambda_{i},\lambda_{i+1})$. Hence, one
obtains

$(\lambda_{i},\lambda_{i+1})\cap\sigma(\hat{\mathfrak{h}}^{(m)})=\emptyset.$
(198)

∎

##  Appendix F The effective dynamics

The effective Hamiltonian constraint reads

$H=p_{\varphi}^{2}-\frac{4\pi}{GL_{0}^{2}\gamma^{2}}\left(\frac{2}{\delta_{b}\delta_{c}}p_{b}\sin(\delta_{b}b)p_{c}\sin(\delta_{c}c)+\frac{1}{\delta_{b}^{2}}p_{b}^{2}\sin^{2}(\delta_{b}b)+\gamma^{2}p_{b}^{2}\right)=:p_{\varphi}^{2}-\frac{4\pi}{GL_{0}^{2}\gamma^{2}}\mathfrak{h}.$
(199)

As $p_{c}\sin(\delta_{c}c)=\gamma mL_{0}\delta_{c}$ is a constant of motion,
it is sufficient to consider the following Hamiltonian constraint for the
evolution of $p_{b}$,

$H^{(m)}=p_{\varphi}^{2}-\frac{4\pi}{GL_{0}^{2}\gamma^{2}}\left(\frac{2\gamma
mL_{0}}{\delta_{b}}p_{b}\sin(\delta_{b}b)+\frac{1}{\delta_{b}^{2}}p_{b}^{2}\sin^{2}(\delta_{b}b)+\gamma^{2}p_{b}^{2}\right)=:p_{\varphi}^{2}-\frac{4\pi}{GL_{0}^{2}\gamma^{2}}\mathfrak{h}^{(m)}.$
(200)

The evolution of $y:=p_{b}\sin(\delta_{b}b)$ with respect to $\varphi$ is
given by

$\displaystyle\frac{{\rm d}y}{{\rm
d}\varphi}=\frac{\sqrt{4\pi}}{\sqrt{G}L_{0}\gamma}\\{y,\sqrt{\mathfrak{h}^{(m)}}\\}=\frac{4\pi}{GL_{0}^{2}\gamma^{2}}\frac{1}{p_{\varphi}}p_{b}^{2}\gamma^{3}\delta_{b}\cos(\delta_{b}b)=\pm\frac{4\pi\delta_{b}\gamma}{GL_{0}^{2}}\frac{1}{p_{\varphi}}\sqrt{p_{b}^{4}-p_{b}^{2}y^{2}}.$
(201)

The Hamiltonian constraint $H^{(m)}=0$ can also be written as

$p_{b}^{2}=-\frac{y^{2}}{\gamma^{2}\delta_{b}^{2}}-\frac{2L_{0}my}{\gamma\delta_{b}}+\frac{GL_{0}^{2}p_{\varphi}^{2}}{4\pi}$
(202)

which, together with (201), gives

$\frac{{\rm d}y}{{\rm
d}\varphi}=\pm\frac{4\pi\sqrt{1+\gamma^{2}\delta_{b}^{2}}}{GL_{0}^{2}\gamma\delta_{b}}\frac{1}{p_{\varphi}}\sqrt{(y-y_{-})(y-y_{-}^{\prime})(y-y_{+}^{\prime})(y-y_{+})}$
(203)

with

$\displaystyle y_{\pm}$
$\displaystyle=L_{0}\gamma\delta_{b}m\left(-1\pm\sqrt{1+\frac{Gp_{\varphi}^{2}}{4\pi
m^{2}}}\right)$ (204) $\displaystyle y_{\pm}^{\prime}$
$\displaystyle=\frac{L_{0}\gamma\delta_{b}m}{1+\gamma^{2}\delta_{b}^{2}}\left(-1\pm\sqrt{1+\frac{Gp_{\varphi}^{2}(1+\gamma^{2}\delta_{b}^{2})}{4\pi
m^{2}}}\right).$

Moreover, according to (202), the maximal value of $p_{b}$ along its dynamical
trajectory is

$(p_{b}^{\rm max})^{2}=L_{0}^{2}m^{2}(1+\frac{Gp_{\varphi}^{2}}{4\pi m^{2}})$
(205)

with which $y_{\pm}$ and $y_{\pm}^{\prime}$ can be rewritten as

$\displaystyle y_{\pm}$
$\displaystyle=L_{0}\gamma\delta_{b}m\left(-1\pm\frac{p_{b}^{\rm
max}}{L_{0}m}\right)$ (206) $\displaystyle y_{\pm}^{\prime}$
$\displaystyle=\frac{L_{0}\gamma\delta_{b}m}{1+\gamma^{2}\delta_{b}^{2}}\left(-1\pm\sqrt{(1+\gamma^{2}\delta_{b}^{2})\frac{(p_{b}^{\rm
max})^{2}}{L_{0}^{2}m^{2}}-\gamma^{2}\delta_{b}^{2}}\right).$

Eq. (206) can be used to fix the dynamical solution by the initial data of
$p_{b}^{\rm max}$.

Solution to (203) is

$\varphi=\pm\frac{GL_{0}^{2}\gamma\delta_{b}}{4\pi\sqrt{1+\gamma^{2}\delta_{b}^{2}}}p_{\varphi}\int_{y_{a}}^{y_{b}}\frac{{\rm
d}y}{\sqrt{(y-y_{-})(y-y_{-}^{\prime})(y-y_{+}^{\prime})(y-y_{+})}}$ (207)

As $p_{b}^{2}=-(y-y_{-})(y-y_{+})\geq 0$ and $y_{-}\leq y_{-}^{\prime}\leq
y_{+}^{\prime}\leq y_{+}$, we have

$y_{a}\in[y_{-}^{\prime},y_{+}^{\prime}],\
y_{b}\in[y_{-}^{\prime},y_{+}^{\prime}].$ (208)

Moreover, because (202) can be written as

$p_{b}^{2}=-\frac{1}{\gamma^{2}\delta_{b}^{2}}(y-y_{-})(y-y_{+}),$ (209)

(208) implies that $p_{b}$ cannot reach $0$ and will bounce at

$p_{b}^{(\pm)}=-(y_{\pm}^{\prime}-y_{-})(y_{\pm}^{\prime}-y_{+}).$ (210)

In order to calculate the integral (207), we define

$t(y)=\sqrt{\frac{y_{+}^{\prime}-y_{-}}{y_{+}^{\prime}-y_{-}^{\prime}}}\sqrt{\frac{y-y_{-}^{\prime}}{y-y_{-}}}.$
(211)

Then (207) becomes

$\displaystyle\varphi=\pm\frac{GL_{0}^{2}\gamma\delta_{b}}{2\pi\sqrt{1+\gamma^{2}\delta_{b}^{2}}}p_{\varphi}\sqrt{\frac{1}{(y_{+}^{\prime}-y_{-})(y_{+}-y_{-}^{\prime})}}\int_{t_{a}}^{t_{b}}\frac{{\rm
d}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}.$ (212)

By choosing the initial data $\varphi_{0}=\varphi(y_{-}^{\prime})$, we finally
have

$\varphi(y)-\varphi_{0}=\pm\frac{GL_{0}^{2}\gamma\delta_{b}}{2\pi\sqrt{1+\gamma^{2}\delta_{b}^{2}}}p_{\varphi}\sqrt{\frac{1}{(y_{+}^{\prime}-y_{-})(y_{+}-y_{-}^{\prime})}}F(\arcsin(t(y))|k),$
(213)

where $F(x|k)$ is the elliptic integral of the first type.

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