††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: Author to whom the
correspondence should be addressed.

# Thermo-induced motional Stark effect on magnetoexciton energy spectra in
monolayer transition-metal dichalcogenides

Duy-Nhat Ly<EMAIL_ADDRESS>Computational Physics Key Laboratory K002,
Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh
City 72759, Vietnam  Dai-Nam Le<EMAIL_ADDRESS>Department of Physics,
University of South Florida, Tampa, FL 33620, United States of America  Atomic
Molecular and Optical Physics Research Group, Advanced Institute of Materials
Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam  Ngoc-Hung Phan
Computational Physics Key Laboratory K002, Department of Physics, Ho Chi Minh
City University of Education, Ho Chi Minh City 72759, Vietnam  Van-Hoang Le
<EMAIL_ADDRESS>Computational Physics Key Laboratory K002, Department of
Physics, Ho Chi Minh City University of Education, Ho Chi Minh City 72759,
Vietnam

###### Abstract

Although energy spectra of an exciton in monolayer transition-metal
dichalcogenides (TMDCs) with a magnetic field were measured at different
temperatures, the theoretical calculations have been performed, neglecting the
temperature effect till now. In the present work, we suggest a new mechanism
that allows the thermal motion of the exciton’s center of mass to affect the
exciton energies. We first recall that the center of mass (c.m.) of a two-body
system in a magnetic field is no longer free and then use a non-trivial scheme
for its separation. The obtained equation for the relative motion contains a
specific term (related to the motional Stark effect) proportional to the c.m.
pseudomomentum, whose average, in turn, is related to the temperature of the
exciton gas. By solving the Schrödinger equation without omitting the motional
Stark potential, we show that the thermo-induced effect on the exciton
energies is significant enough for experimental detection. The symmetry
breaking of the system due to the additional term and the thermo-induced
effect on the exciton radius and diamagnetic coefficient are also discussed by
examining the wave functions. Some consequences are studied, especially the
enhancement of exciton lifetime and the appearance of new peaks in the
absorption spectra besides those of the s-states. This study could fill the
gap between the present theoretical calculations and experimental measurements
on energy spectra of magnetoexciton in monolayer TMDCs at finite temperatures.

exciton, transition-metal dichalcogenides, finite temperature, motional Stark
effect potential, thermo-induced effect

##  I Introduction

During the last two decades, monolayer transition-metal dichalcogenides
(TMDCs) have become hot spots for studying the formation of excitons because
of their unique property of electron-hole interaction [1, 2, 3, 4]. Unlike
three-dimensional exciton in novel semiconductors, neutral exciton in these
monolayer TMDCs is thermally stable at room temperature, even for the Rydberg
states. Their high binding energy provides beneficial optical properties in
both UV and IR ranges [5, 6, 7, 8, 9]. Especially, exciton energy spectra
under the presence of a constant magnetic field have been of great interest
recently both in experimental observations and theoretical studies due to
their incredible potential in retrieving several physical quantities of
monolayer TMDCs, such as effective masses of electron and hole, the $g-$Landé
factors, and 2D polarizability [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
Therefore, it is essential to qualitatively comprehend the influences of
external factors like the electric field, deformation, or temperature on the
energy spectra of magnetoexciton in the monolayer TMDCs.

To do so, we need to solve the Schrödinger equation of the electron-hole pair
in the framework of the effective mass approximation. One required task before
solving the Schrödinger equation is to separate the movement of the electron-
hole mass center. However, for a two-body system such as an exciton in a
magnetic field, center dynamics can not be removed totally from the equation
describing the electron-hole relative motion. The variable-separation
procedure for this system is no longer trivial because the magnetic field
breaks the system’s translation symmetry, which leads to the nonconservation
of the total momentum of the electron-hole pair [21, 22, 23]. Fortunately,
there is instead another constant of motion named pseudomomentum, which
remains in the relative motion Hamiltonian via a term influencing the exciton
in the same way as the Stark effect; that is why this term is called motional
Stark potential [24, 25]. Recent theoretical studies of magnetoexciton in
monolayer TMDCs [13, 17, 20] always neglect this thermo-induced potential that
is reasonable only at zero temperature. However, the experimental observations
were conducted not only at low temperatures [9, 10, 12, 15, 14, 18, 19, 16]
but also at room temperature [5, 6, 7, 8, 9]. Hence, we believe that the
motional Stark effect arising from the thermal fluctuation of the c.m.
pseudomomentum should be considered when calculating the energy spectra of
magnetoexciton in monolayer TMDCs.

It is well-known that the electron-hole recombination and, as well as,
exciton-phonon scattering at finite temperatures affect the imaginary part of
exciton energy that makes exciton decay [26, 27, 28, 29, 3, 30, 31, 32].
However, the motional Stark potential, linearly proportional to the magnetic
intensity, is inconspicuous compared to the diamagnetic potential depending on
the squared magnetic intensity and may contribute only to the energy shift but
not to the exciton lifetime. Nevertheless, the thermo-induced Stark potential
could make the average radius of the electron-hole pair larger and
consequently enhance the radiative lifetime of excitons in monolayer TMDCs
[27, 31, 32]. Besides, the wave functions can be affected by the symmetry
breaking of the system due to the additional thermo-induced term. Therefore,
we aim to examine the effect of finite temperature on the exciton energy
spectra, the diamagnetic coefficient, and the exciton radius as a function of
magnetic intensity and their consequence on the absorption spectra and the
exciton radiative lifetime.

The plan of this paper is as follows. Section II presents the theoretical
background for our study, including the separation of the c.m. motion, the
introduction of the motional Stark potential as well as the assumption about
its temperature dependence, and the numerical method for solving the
Schrödinger equation of relative motion between electron and hole. Then the
thermo-induced motional Stark effect on energy spectra, the diamagnetic
coefficients, and the electron-hole average distances are given in Section
III. The symmetry-breaking effect on the absorption spectra is also examed in
Section III. Section IV encloses our conclusion and further discussion.

Figure 1: Exciton having conserved thermal pseudomomentum in a monolayer TMDC
with a magnetic field. The interaction between electron and hole is described
by the Rytova-Keldysh potential $V_{RK}(r)$.

##  II Theoretical background

###  II.1 The Schrödinger equation

Within the effective mass approximation, neutral exciton in a monolayer TMDC
is described as a two-dimensional system of one electron and one hole
interacting with each other by the potential
${\hat{V}}_{h-e}(|{\mathbf{r}_{h}}-{\mathbf{r}_{e}}|)$ in the $Oxy$ plane, as
shown in Fig. 1. In the presence of a constant magnetic field
$\mathbf{B}=B\mathbf{e}_{z}$, perpendicular to the system’s plane, the
effective mass Hamiltonian of the neutral exciton system has the form

$\displaystyle{\hat{H}}_{X}=\frac{{\hat{\mathbf{p}}_{h}}^{2}}{{{2{m^{*}_{h}}}}}+\frac{{\hat{\mathbf{p}}_{e}}^{2}}{{{2{m^{*}_{e}}}}}+\frac{{eB}}{{2{m^{*}_{e}}}}{\hat{l}_{ez}}-\frac{{eB}}{{2{m^{*}_{h}}}}{\hat{l}_{hz}}$
$\displaystyle+\frac{{{e^{2}}{B^{2}}}}{{8{m^{*}_{h}}}}\mathbf{r}_{h}^{2}+\frac{{{e^{2}}{B^{2}}}}{{8{m^{*}_{e}}}}\mathbf{r}_{e}^{2}+{\hat{V}}_{h-e}(|{\mathbf{r}_{h}}-{\mathbf{r}_{e}}|)\,.$
(1)

Here, $m^{*}_{e}$ and $m^{*}_{h}$ are the effective masses of the electron and
hole; $e$ is the elementary charge with the positive value. The symbols
$\hat{\mathbf{p}},\hat{\mathbf{r}}$, and $\hat{l}_{z}$ denote the momentum,
coordinate, and angular momentum operators, respectively, with the indices $e$
and $h$ for the electron and hole.

In the c.m. and relative coordinates

$\displaystyle\mathbf{R}=\frac{m^{*}_{h}\mathbf{r}_{h}+m^{*}_{e}\mathbf{r}_{e}}{m^{*}_{e}+m^{*}_{h}}\,,\quad\mathbf{r}=\mathbf{r}_{e}-\mathbf{r}_{h}\,,$
(2)

Hamiltonian (II.1) can be rewritten as

$\displaystyle{\hat{H}}_{X}=$
$\displaystyle\frac{\hat{\mathbf{p}}^{2}}{{2\mu}}+\frac{1-\sigma}{1+\sigma}\,\frac{{eB}}{2\mu}{{\hat{l}}_{z}}+\frac{M-3\mu}{\mu
M}\frac{{{e^{2}}{B^{2}}}}{8}{\mathbf{r}^{2}}+{\hat{V}}_{h-e}(\mathbf{r})$ (3)
$\displaystyle+\frac{\hat{\mathbf{P}}^{2}}{{2M}}+\frac{{{e^{2}}{B^{2}}}}{{8\mu}}{\mathbf{R}^{2}}+\frac{1-\sigma^{2}}{\sigma}\frac{{{e^{2}}{B^{2}}}}{4M}\;\mathbf{r}\cdot\mathbf{R}$
$\displaystyle+\frac{e}{2M}(\mathbf{B}\times\mathbf{r})\cdot\hat{\mathbf{P}}+\frac{e}{2\mu}(\mathbf{B}\times\mathbf{R})\cdot\hat{\mathbf{p}}\,,$

with the total mass $M=m^{*}_{e}+m^{*}_{h}$, the exciton reductive mass
$\mu=m^{*}_{e}m^{*}_{h}/(m^{*}_{e}+m^{*}_{h})$, and the ratio of masses
$\sigma=m^{*}_{e}/m^{*}_{h}$. Here, $\hat{\mathbf{p}}$ and $\hat{l}_{z}$ are
the operators of the momentum and angular momentum of the relative motion;
$\hat{\mathbf{P}}$ is the c.m. momentum operator.

It should be noted that the c.m. momentum is not conserved, i. e.
$[{\hat{H}}_{X},\hat{\mathbf{P}}]\neq 0$, and consequently, the Hamiltonian
(3) for the considered system is not variable separable. Fortunately, there
exists a way to extract the relative motion for such a system, as it was given
a long time ago for a more generalized problem in N-dimensional space [22].
Particularly, the variable separation for a three-dimensional hydrogen atom in
a constant magnetic field was presented in [23]. This variable-separation
method was also applied to the case of two-dimensional space considering the
Hamiltonian (3) to get the relative motion of an exciton in a magnetic field
[33, 21]. We will recall the main equations of [21] for our application.

First, we consider the operator
$\hat{\mathbf{P}}_{0}=\hat{\mathbf{P}}-\frac{1}{2}e\mathbf{B}\times\mathbf{r}$
and verify that it commutes with the Hamiltonian (3), i. e.,
$[\hat{\mathbf{P}}_{0},{\hat{H}}_{X}]=0$. Therefore, the pseudomomentum
described by the operator $\hat{\mathbf{P}}_{0}$ is conserved for the
considered system, and we can define the eigenvector and eigenfunction of this
operator as in the following equation

$\hat{\mathbf{P}}_{0}\,{e^{\frac{i}{\hbar}\left({\mathbf{K}+\frac{1}{2}\,e\mathbf{B}\times\mathbf{r}}\right)\cdot\mathbf{R}}}=\mathbf{K}\;{e^{\frac{i}{\hbar}\left({\mathbf{K}+\frac{1}{2}\,e\mathbf{B}\times\mathbf{r}}\right)\cdot\mathbf{R}}}.$
(4)

Consequently, we can define the wave function in the form

$\Psi_{\mathbf{K}}(\mathbf{R},\mathbf{r})={e^{\frac{i}{\hbar}\left({\mathbf{K}+\frac{1}{2}e\mathbf{B}\times\mathbf{r}}\right)\cdot\mathbf{r}}}\,\psi_{\mathbf{K}}(\mathbf{r})$
(5)

for the exciton with the c.m. pseudomomentum ${\mathbf{K}}$. Then, plugging
wave function (5) into the Schrödinger equation with the Hamiltonian (3), we
have an equation for the wave function of the relative motion
$\psi_{\mathbf{K}}(\mathbf{r})$ as follows

$\displaystyle\left\\{\frac{\hat{\mathbf{p}}^{2}}{2\mu}+\frac{1-\sigma}{1+\sigma}\right.\frac{{eB}}{2\mu}{{\hat{l}}_{z}}+\frac{{{e^{2}}{B^{2}}}}{{8\mu}}{\mathbf{r}^{2}}+{\hat{V}}_{h-e}(\mathbf{r})\quad\quad\quad$
$\displaystyle\left.-\frac{e}{M}({\mathbf{B}\times\mathbf{K}})\cdot\mathbf{r}-E\right\\}\psi_{\mathbf{K}}(\mathbf{r})=0.$
(6)

For the exciton in monolayer TMDCs such as $\textrm{WSe}_{2}$ as considered in
this work, the interaction between electron and hole
$\hat{V}_{h-e}(\mathbf{r})$ is described by the Rytova-Keldysh potential [34,
35]:

$V_{RK}\left(r,r_{0},\kappa\right)=-\frac{e^{2}}{8\varepsilon_{0}r_{0}}\left[H_{0}\left(\frac{\kappa\,r}{r_{0}}\right)-Y_{0}\left(\frac{\kappa\,r}{r_{0}}\right)\right],$
(7)

where $H_{0}$ and $Y_{0}$ are the zero-order Struve and Bessel functions of
the second kind; $\kappa$ is the average dielectric constant of the
surrounding material; $\varepsilon$ is the permittivity of vacuum. The
screening length $r_{0}$ has the physical meaning related to the $2D$
polarizability $\chi_{2D}$ by the formula $r_{0}=2\pi\chi_{2D}$ [36, 37, 1,
2].

###  II.2 Thermo-induced motional Stark potential

One essential point of this work is to consider the specific term of equation
(II.1) which has the form

$\displaystyle
V_{mS}(\mathbf{r})=-\frac{e}{M}({\mathbf{B}\times\mathbf{K}})\cdot\mathbf{r}$
(8)

and is related to the motional Stark effect [24, 25]. This term was omitted in
the previous theoretical studies of magnetoexcitons in monolayer TMDCs [14,
19]; however, in the present study, we would like to show its significant
contribution to the exciton energies at room temperature. For this reason, we
will take a quick look at this term at this stage to show its relation to the
exciton gas temperature.

We consider neutral excitons in monolayer TMDCs as a gas system with a low
density (the average distance between two excitons is much larger than the
thermal wavelength) so that the Maxwell-Boltzmann statistics is still valid,
and the formation of trions or biexcitons is negligible. Here, the density
$n_{X}$ is the number of neutral excitons per unit of area, and the thermal
wavelength is $\lambda_{th}=\sqrt{\frac{2\pi\hbar^{2}}{Mk_{B}T}}$. Therefore,
the condition for the validity of the Maxwell-Boltzmann statistics is
$n_{X}\lambda_{th}^{2}\ll 1$, i.e., $n_{X}\ll\frac{Mk_{B}T}{2\pi\hbar^{2}}$.
Practically, for monolayer WSe2, exciton density $n_{X}$ could be modulated
around $10^{11}\text{--}10^{12}\text{ cm}^{-2}$ in real experiments [38, 39,
40, 41], which is much smaller than its limit at room temperature
$\frac{Mk_{B}T}{2\pi\hbar^{2}}=4.3\times 10^{12}\text{ cm}^{-2}$. Since the
pseudomomentum $\hat{\mathbf{P}}_{0}$ has two freedom degrees, we can estimate
its root-mean-square at the temperature $T$ by the equipartition theorem as

$\displaystyle\frac{1}{2M}\left<\hat{P}^{2}_{0}\right>=k_{B}T,$

where $k_{B}$ is the Boltzmann constant.

On the other hand, $\hat{\mathbf{P}}_{0}$ is a constant of motion and has the
eigenvalue $\mathbf{K}$; therefore, we have the relationship
$|K|=\sqrt{2Mk_{B}T}$. The term (8) is now related to the temperature as
follow:

$\displaystyle V_{mS}(\mathbf{r})=-\sqrt{\frac{2k_{B}T}{M}}eB\,x.$ (9)

Here, without losing generality, we consider the case where $\mathbf{K}$ is
along the $Oy$ axis so that vector $\mathbf{B}\times\mathbf{K}$ is along the
$Ox$ axis.

The term $V_{mS}$ depends on the magnetic field and exists only with finite
temperatures $T>0$. Therefore, we call it the thermo-induced motional Stark
potential. For illustration, we show the effective potential of equation
(II.1) in Fig. 2 for two cases, with and without the thermo-induced motional
Stark potential, i.e., $T\neq 0$ and $T=0$. This figure clearly shows that at
room temperature, the contribution of the added term on the effective
potential is noticeable for the 2s and higher Rydberg states, and it would
influence the exciton energy spectra.

Figure 2: The effective potentials for relative motion in two cases: $T=0K$
(black) and $T=300K$ (red), i.e., without and with the motional Stark term.
The difference between the potentials is noticeable for $2s$ and higher
Rydberg states, where the exciton radius is larger than 6.5 nm.

Especially, the effective potential in Fig. 2 shows no tunneling effect, even
considering the thermo-induced motional Stark potential. That means the
exciton is always in its bound states, unlike the well-known LoSurdo-Stark
effect in the two-dimensional electron gas [42, 43]. Indeed, this fact can be
understood from the thermo-induced motional Stark potential (9). This part is
linearly proportional to the variable $x$; however, the main part of the
effective potential, the diamagnetic potential
$V_{diamag}=\frac{e^{2}B^{2}}{8\mu}\mathbf{r}^{2}$, is quadratic proportional
to the electron-hole distance and consequently dominant at the large
separation of the electron-hole pair that keeps the exciton in bound states.
Instead of tunneling, we believe the thermo-induced motional Stark potential
could cause the Stark shift in the energy spectra, as we will show in Section
III.

###  II.3 Method of solving the Schrödinger equation

The Schrödinger equation (II.1) can be solved by various methods. In this
work, we use the standard perturbation theory to predict the main rules of the
thermo-induced motional Stark effect. Then, we also apply the
Feranchuk–Komarov (FK) operator method, a numerical method suggested in
Refs.[44, 17] for an exciton in monolayer TMDCs in a magnetic field, to get
highly accurate exciton energies for a more detailed analysis. For our
purpose, we recall here some main points of the FK operator method. It should
be noted that the included thermo-induced motional Stark potential does not
lead to any additional difficulty in calculations.

First, the Schrödinger equation (II.1) is rewritten in the atomic units where
the energy $E$ and the coordinates $x,y$ are given in the effective Hartree
unit $2Ry^{*}=\mu e^{4}/16\pi^{2}\varepsilon_{0}^{2}\hbar^{2}$ and the
effective Bohr radius $a_{0}^{*}=4\pi\varepsilon_{0}\hbar^{2}/\mu e^{2}$,
respectively; the dimensionless magnetic intensity $\gamma$ is related to the
magnetic field by the equation $B=2\gamma\times\mu\hbar Ry^{*}/e$. Next, the
Levi-Civita transformation [45]

$x=u^{2}-v^{2},\;y=2uv$ (10)

is applied to the dimensionless Schrödinger equation to rewrite it in the
$(u,v)$ space as

$\displaystyle\left\\{-\frac{1}{8}\left(\frac{\partial^{2}}{\partial{u}^{2}}+\frac{\partial^{2}}{\partial{v}^{2}}\right)+\left(\frac{1-\sigma}{1+\sigma}\frac{\gamma}{2}\,{\hat{l}}_{z}-E\right)\left({u}^{2}+{v}^{2}\right)\right.$
$\displaystyle\quad\quad+\frac{\gamma^{2}}{8}\left({u}^{2}+{v}^{2}\right)^{3}+\frac{\sqrt{2\sigma\,\theta}}{1+\sigma}\gamma\left(u^{4}-v^{4}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.+\,V(u,v)\right\\}\psi(u,v)=0,$
(11)

where $\theta=k_{B}T$ is dimensionless temperature; and
$V(u,v)=\left({u}^{2}+{v}^{2}\right)V_{RK}$. For convenience in calculations,
we rewrite the potential $V(u,v)$ in the Laplace transform [46] as

$V(u,v)=-\frac{1}{\kappa}\int\limits_{0}^{+\infty}\frac{dq}{\sqrt{1+\alpha^{2}q^{2}}}\;\textrm{e}^{-q(u^{2}+v^{2})}(u^{2}+v^{2}),$
(12)

where the dimensionless parameter $\alpha=r_{0}/\kappa a^{*}_{0}$ is used
instead of the screening length $r_{0}$. In equations (II.3) and (12), we have
used the formulas

$r=u^{2}+v^{2},\quad\quad{\hat{l}_{z}}=-\frac{i}{2}\left(v\frac{\partial}{\partial
u}-u\frac{\partial}{\partial v}\right)$ (13)

obtained by the Levi-Civita transformation (10) for distance $r$ and angular
momentum $\hat{l}_{z}$.

Equation (II.3) is similar to those describing a two-dimensional anharmonic
oscillator, suggesting we use the harmonic oscillator wave functions as a
basis set. Moreover, the algebraic approach based on annihilation and creation
operators can be utilized for calculations. We use the notation

${|k,m,\omega\rangle}$ (14)

for the wave vector of the two-dimensional harmonic oscillator in the $(u,v)$
space with the angular frequency $\omega$, where $k$ is the principal quantum
number ($k=0,1,2,...$), and $m$ is the magnetic quantum number ($-k\leq m\leq
k$). We note that the frequency $\omega$ can be regarded as a free parameter
and will choose its value appropriate for regulating the convergence rate of
the numerical solutions.

We use the wave vectors (14) with the running indices $k$ and $m$ as a basis
set to expand the solution of the Schrödinger equation (II.3) as

${|\psi^{(s)}(\omega)\rangle}=\sum_{k=0}^{s}\sum_{m=0}^{k}C_{km}^{(s)}{|k,m,\omega\rangle}$
(15)

with $N=(s+1)(s+2)/2$ unknown coefficients $C_{km}^{(s)}$ needed to define. We
note that equation (II.3) is invariant under the transformation
$x\rightarrow-x,\,\mathbf{B}\rightarrow-\mathbf{B}$; therefore, we can use
only $m\geq 0$ in the expansion (15). In the wave function (15), we use only
$(s+1)/(s+2)/2$ basis set functions so that the number $s$ can be regarded as
an approximation order of the solutions. In practice, we will increment the
$(s)$-order until getting the desired precision.

Plugging the wave function (15) into equation (II.3) and acting to the left
with $\langle j,m^{\prime}|$, we lead this equation to $N$ linear equations
for the coefficients $C_{km}^{(s)}$ and the corresponding energy ${E}^{(s)}$.
All the coefficients of the equations can be calculated analytically. We also
can rewrite the linear equations into the $N\times N-$ matrix eigenvalue
equation, where the eigenvalue is ${{E}}^{(s)}$, and the eigenvector contains
$N$ elements $C^{(s)}_{km}$. The matrix eigenvalue equation can be solved by
the subroutine dsygvx.f of the Linear Algebra Package (LAPACK) [47].

##  III  Thermo-induced motional Stark effect

###  III.1 Asymptotic behaviour of energy spectra in low and high magnetic
field

To see how the thermo-induced motional Stark potential affects the asymptotic
behaviors of exciton energies in the limits of low and high magnetic
intensities, we calculate the energies analytically by applying the
perturbation theory to the Schrödinger equation (II.1). The obtained general
rules are presented here and will be verified by a more comprehensive analysis
in Section III.2 by the numerical method given in Section II.3.

For low magnetic intensity, restricted by the condition
$l_{B}=\sqrt{{\hbar}/{eB}}\gg\left<r\right>_{nm}$, we have the energy of the
($n,m$) quantum state in the second perturbation order as

$\displaystyle
E^{(2)}_{nm}(B,T)=E_{nm}^{(0)}+\frac{1-\sigma}{1+\sigma}\frac{m\hbar}{2\mu}\,eB\quad\quad$
$\displaystyle+\frac{\left<r^{2}\right>_{nm}}{8\mu}\,e^{2}B^{2}-\alpha_{nm}\frac{k_{B}T}{M}\,e^{2}B^{2},$
(16)

where $E^{(0)}_{nm}$, $\left<r^{2}\right>_{nm}$, and $\alpha_{nm}$ are the
zero-field energy, squared radius, and polarizability of the exciton. To
calculate these quantities, we need to solve the Schrödinger equation in the
zeroth order of approximation. However, we do it another way by numerical
solving the equation (II.3) and fitting the obtained results to the formula
(III.1). The concrete values are presented in Table 1 for $1s$, $2s$, $3s$,
$2p_{-1}$, and $2p_{+1}$ states.

Table 1:  The zero-field energy, squared radius, and polarizability for the exciton quantum states $1s$, $2s$, $3s$, $2p_{-1}$, and $2p_{+1}$. | $1s$ | $2s$ | $3s$ | $2p_{-1}$ | $2p_{+1}$
---|---|---|---|---|---
$E^{(0)}_{nm}$ (eV) | -0.1686 | -0.0386 | -0.0166 | -0.0498 | -0.0498
$\left<r^{2}\right>_{nm}$ (nm2) | 2.63 | 45.8 | 241 | 21.1 | 21.1
$\alpha_{nm}$ (nm2/eV) | 3.91 | -112 | -6060 | 259 | 240

We can see that the last term in the exciton energy (III.1), the motional
Stark correction, is quadratic proportional to the magnetic field. That means
the thermo-induced effect also contributes to the diamagnetic coefficient
$\sigma_{nm}$, defined by the equation:

$\displaystyle\sigma_{nm}(T)$ $\displaystyle=\lim\limits_{B\to
0}\frac{1}{2}\frac{\partial^{2}E_{nm}(B,T)}{\partial B^{2}}$ (17)
$\displaystyle=\frac{e^{2}}{8\mu}\left<r^{2}\right>_{nm}-\frac{e^{2}}{M}\alpha_{nm}k_{B}T.$

Table 2 shows some values for the zero-temperature diamagnetic coefficient
$\sigma^{0}_{nm}=\frac{e^{2}}{8\mu}\left<r^{2}\right>_{nm}$ and their thermo-
induced corrections $\Delta\sigma_{nm}=-\frac{e^{2}}{M}\alpha_{nm}k_{B}T$ for
the $1s$, $2s$, $3s$, $2p_{-1}$, and $2p_{+1}$ states at room temperature. The
thermo-induced corrections are about $8\%$ for $1s$, $13\%$ for $2s$, and
extremely high for higher states: $2p_{+1}$, $2p_{-1}$, and $3s$ ($56\%$,
$60\%$, and $130\%$). These are significant enough to impact experimental
measurement that we should pay attention to when measuring exciton energies at
finite temperatures.

Table 2:  The zero-temperature diamagnetic coefficients and their thermo-induced corrections (at room temperature) in unit of $\mu$eV/Tesla2 for quantum states $1s$, $2s$, $3s$, $2p_{-1}$, and $2p_{+1}$. The experimental data are picked up from Ref. [14]. | $1s$ | $2s$ | $3s$ | $2p_{-1}$ | $2p_{+1}$
---|---|---|---|---|---
$\sigma_{nm}$ [14] | $0.31\pm 0.02$ | $4.6\pm 0.2$ | $22\pm 2$ | - | -
$\sigma^{0}_{nm}$ | 0.289 | 5.039 | 26.53 | 2.429 | 2.429
$\Delta\sigma_{nm}$ | -0.022 | 0.637 | 34.42 | -1.469 | -1.361
$\frac{|\Delta\sigma_{nm}|}{\sigma^{0}_{nm}}$ | 8% | 13% | 130% | 60% | 56%

For high magnetic intensity, restricted by condition
$l_{B}=\sqrt{\hbar/eB}\ll\left<r\right>_{nm}$, the harmonic oscillator
potential becomes dominant compared to the Rytova-Keldysk potential. In this
case, the main part of the Hamiltonian is the magnetic term; therefore, we can
obtain the Landau levels for energies with the thermo-induced Stark
corrections by the perturbation theory as

$\displaystyle E^{(2)}_{nm}(B,T)=$
$\displaystyle\frac{\hbar}{2\mu}\left(2n-|m|-1+\frac{1-\sigma}{1+\sigma}m\right)eB$
(18) $\displaystyle\quad\quad-\frac{8\mu\beta_{nm}}{M}k_{B}T,$

where the dimensionless coefficients $\beta_{nm}$ are independent of the
temperature $T$. Because of the neglect of Coulomb interaction, the energy
formula (18) is valid only for very high magnetic intensity, extremely higher
than the laboratory limit of about 150 Tesla. Nevertheless, this formula gives
the right rule for thermo-induced shifts of the Landau levels $\Delta
E=-\frac{8\mu\beta_{nm}}{M}k_{B}T$, i.e., linearly proportional to the
temperature and almost independent of the magnetic field.

###  III.2 Effect on energy spectra of magnetoexciton

After getting the general analytical rules of how the thermo-induced motional
Stark potential affects the exciton energies and, consequently, the
diamagnetic coefficients by the perturbation theory, we now investigate this
effect in detail by the numerical method presented in Subsection II.3.

We will calculate exciton energies for the specific case of $\textrm{WSe}_{2}$
where the parameters in equations (II.1) and (7) have the following values:
$\kappa=4.5$, $r_{0}=4.21$, $\mu=0.20\times m_{e}$, $\sigma=0.94$,
$M=0.80\times m_{e}$, taken from Refs. [14, 19]. Here, $m_{e}$ is the electron
mass. Contrary to the previous studies, we calculate energies not only for the
temperature of 0 K but also 300K and compare the results to demonstrate the
thermo-induced motional Stark effect.

Table 3 shows the energies of $1s$, $2s$, and $3s$ states. We can see that the
temperature does not influence the $1s$ state much, while the thermo-induced
effect is noticeable for the $2s$ and higher Rydberg states for the magnetic
intensity of more than 60 Tesla. Particularly for the magnetic intensity of 90
Tesla, the energy difference $E_{2s}$ (300 K) $-E_{2s}$ (0 K) is about 1.5 meV
while $E_{3s}$ (300 K) $-E_{3s}$ (0 K) is about 3.9 meV. These differences are
big enough compared to the experimental sensitivity of 1 meV. For an
illustration of the effect, we also show in Fig. 3 the transition energies
$\Delta E_{2s-1s}=E_{2s}-E_{1s}$ and $\Delta E_{3s-1s}=E_{3s}-E_{1s}$
depending on the magnetic intensity at 0 K and 300 K. It is clear that for the
$3s$ state, the temperature effect is significant even for the median values
of the magnetic field.

We also consider the $2p_{-1}$ and $2p_{+1}$ states and present their energies
at temperatures 0 K and 300 K in Table 4. We can see that the temperature
effect is also noticeable. Particularly for the magnetic intensity of 90
Tesla, the differences in energies at two different temperatures are 4.5 meV
for $2p_{-1}$ and 1.9 meV for $2p_{+1}$, experimentally sensitive. To
demonstrate the effect, we plot the separation energy $\Delta
E_{p}=E_{p_{+1}}-E_{p_{-1}}$ as a function of magnetic intensity at three
temperatures: 0 K, 200 K, and 300 K in Fig. 4.

Table 3: Energies of $1s,\,2s,\,\text{and}\,3s$ states (eV) with different magnetic intensity $B$ (Tesla) at temperature 0 K and 300 K for $r_{0}$ = 4.2086 nm, $\mu$ = 0.2039 $m_{e}$, $\kappa=4.5$. | $E_{1s}$ |  | $E_{2s}$ |  | $E_{3s}$
---|---|---|---|---|---
$B\;(T)$ | 0 K | 300 K |  | 0 K | 300 K |  | 0 K | 300 K
0 | -0.168552 | -0.168552 |  | -0.038554 | -0.038554 |  | -0.016552 | -0.016552
1 | -0.168552 | -0.168552 |  | -0.038549 | -0.038548 |  | -0.016526 | -0.016491
10 | -0.168524 | -0.168526 |  | -0.038066 | -0.038003 |  | -0.014204 | -0.012149
30 | -0.168298 | -0.168318 |  | -0.034507 | -0.034043 |  | -0.001283 | 0.002143
60 | -0.167544 | -0.167621 |  | -0.024701 | -0.023575 |  | 0.025947 | 0.029682
90 | -0.166316 | -0.166480 |  | -0.011613 | -0.010125 |  | 0.057185 | 0.061075
120 | -0.164649 | -0.164920 |  | 0.003552 | 0.005150 |  | 0.090569 | 0.094353
Figure 3: The transition energies $\Delta E_{2s-1s}$ (a) and $\Delta E_{3s-1s}$ (b) depending on the magnetic intensity $B$ at temperatures 300 K and 0 K. Table 4: Energies of $2p_{-1}$ and $2p_{+1}$ states (eV) with different magnetic intensities $B$ (Tesla) at temperatures 0 K and 300 K for $r_{0}$ = 4.2086 nm, $\mu=0.2039~{}m_{e}$, $\kappa=4.5.$ | $E_{2p_{-1}}$ |  | $E_{2p_{+1}}$
---|---|---|---
$B\;(T)$ | 0 K | 300 K |  | 0 K | 300 K
0 | -0.049782 | -0.049782 |  | -0.049782 | -0.049782
1 | -0.049787 | -0.049789 |  | -0.049771 | -0.049772
10 | -0.049627 | -0.049806 |  | -0.049463 | -0.049554
30 | -0.047999 | -0.049363 |  | -0.047508 | -0.048001
60 | -0.043008 | -0.046259 |  | -0.042027 | -0.043234
90 | -0.035959 | -0.040457 |  | -0.034487 | -0.036335
120 | -0.027536 | -0.032789 |  | -0.025574 | -0.027961
Figure 4: The separation energy $\Delta E_{2p}=E_{2p_{+1}}-E_{2p_{-1}}$
depending on the magnetic intensity $B$ at temperatures 0 K, 200 K, and 300 K.

###  III.3 Breaking the system’s symmetry and its consequences

It is well-known that the system of a two-dimensional exciton in a magnetic
field, perpendicular to the monolayer TMDC plane, has the potential energy
dependent on radius $r=\sqrt{x^{2}+y^{2}}$ only and consequently possesses the
SO(2) symmetry. However, if included, the thermo-reduced motional Stark
potential (9), which depends on the angle as $\sim r\cos{\varphi}$, will break
this symmetry. In this case, the angular momentum $l_{z}$ is not conserved,
and the magnetic number $m$ is not a good quantum number. The consequence is
that there are no true s-states anymore but only mixed states with $m\neq 0$
from the basis functions.

Indeed, the symmetry breaking is clear in the squared modulus of the wave
functions $|{\psi}|^{2}$ of $2s$ and $3s$ states at temperatures of 0K and
300K, as shown in Fig. 5. The effect is also seen in the wave function of the
$1s$ state but is very slight. We also demonstrate the deformation of the wave
functions $2p_{-1}$ and $2p_{+1}$ by the thermo-induced motional Stark
potential in Fig. 6.

Figure 5: The squared modulus of the wave functions $|\psi|^{2}$ of $1s$,
$2s$, and $3s$ states at temperatures of 0K and 300 K for the magnetic field
of 90 Tesla. For color clarity, the wave functions are normalized so that the
highest value of the modulus equals 1. The normalization factors $F$ are
indicated in the figure. Figure 6: As Fig.5 but for the $2p_{-1}$ and
$2p_{+1}$ states.

The deformation of the wave functions leads to the change of the exciton
radius. Indeed, Figure 7 shows the exciton radii depending on the magnetic
intensity for $1s$, $2s$, and $3s$ states at two temperatures, 0 K and 300 K,
and the thermo-induced effect is noticeable. For example, for the magnetic
intensity of 90 Tesla, the ratio $\Delta\left<r\right>/\left<r\right>$ has the
following values: $0.4\%$ for $1s$, $1.4\%$ for $2s$, and $0.6\%$ for $3s$.

This change is essential for the exciton radiative lifetime $\tau_{rad}$,
which is related to the average electron-hole distance $\left<r\right>$ by the
following scaling law [26, 32]:

$\tau_{rad}\sim\left<r\right>^{\xi}.$ (19)

The scaling factor $\xi$ mainly depends on the dimensionality of the exciton.
We roughly take the value of $\xi\approx 3.5$ extracted by studying exciton in
hBN, diamond, and silicon provided in Ref. [32].

Figure 7: Exciton radius depending on the magnetic field at temperatures of 0
K and 300 K for $1s$, $2s$, and $3s$ states.

To have an analytical estimation, we get the formula for the exciton radius of
s-states by the perturbation theory in the low magnetic field limit as

$\left<r\right>^{T}=\left<r\right>+g\,e^{2}B^{2}\frac{k_{B}T}{M},$ (20)

where the radius $\left<r\right>^{T}$ is calculated using the wave functions
with thermo-induced effect, while $\left<r\right>$ is calculated by the free-
field wave functions (no thermo-induced effect). The coefficient $g$ has the
following values $0.17\times 10^{-3}$, $0.04$, and $-7.5$ (nm3/meV2) for $1s$,
$2s$, and $3s$ states, respectively. Plugging formula (20) into equation (19),
we can get the change of the exciton radiative lifetime $\Delta\tau_{rad}$ as

$\frac{\Delta\tau_{rad}}{\tau_{rad}}=\xi\,\frac{\Delta\left<r\right>}{\left<r\right>},$
(21)

where $\Delta\left<r\right>=ge^{2}B^{2}\frac{k_{B}T}{M}$ for the low magnetic
intensity. For high magnetic field the correction $\Delta\left<r\right>$
should be calculated numerically. We estimate the ratio (21) for the $1s$,
$2s$, and $3s$ states at room temperature and magnetic field of $90$ Tesla and
get the following values: $1.4\%$, $4.9\%$, and $2.1\%$, which are
considerable. The thermo-induced correction to the radiative lifetime (21)
qualitatively agrees with both first-principle calculations and experimental
observations in Refs. [28, 29, 31] for the $1s$ state, meaning that it always
enhances the lifetime and is linearly proportional to the temperature. For the
$2s$ state, the lifetime is also enhanced by the thermal motion, but for the
$3s$ state, the lifetime is reduced according to our calculation. Therefore,
this effect (enhancement/reduction of the lifetime) needs an additional
investigation.

###  III.4 Thermal effect on absorption spectra of magnetoexciton

According to the linear response theory of exciton, the magneto-excitonic
absorption coefficient can be calculated from exciton energy spectra and
radiative lifetime via the Elliott formula [48, 49, 50, 51] as

$\displaystyle\alpha(\omega)=C\,\text{Im}\sum_{n,m}\frac{\omega\,\left|\psi_{nm}(\mathbf{r}=0)\right|^{2}}{E_{nm}+E_{g}-\hbar\omega+i\hbar\tau^{-1}}\,,$
(22)

where $C=\frac{4\pi\left|p_{cv}\right|^{2}}{n_{b}c}$ depends on the background
refractive index of materials $n_{b}$ and dipole matrix element of interband
transition $p_{cv}$. We are interested in the general picture only and thus
choose the coefficient $C$ to normalize the free-field $1s$ peak to $1$ and
then use it as a constant for all other states. In Eq. (22), the bandgap
energy is taken from Ref. [14] by $E_{g}=$ 1.89 eV.

Usually, the total lifetime takes into account both radiative and non-
radiative dephasing effects
$\frac{1}{\tau}=\frac{1}{\tau_{rad}}+\frac{1}{\tau_{non\;rad}}$. However, we
roughly estimate the exciton lifetime $\tau$ by formula (19) and fit the
coefficient from the experiment data [28] for the $1s$ state, which suggests
the exciton lifetime around picosecond.

Figure 8: Normalized optical absorption spectra at temperature of 0 K (a and
b) and at room temperature (c)

Figure 8 illustrates the normalized linear optical absorption spectra for the
quantum states with principal quantum number $n<4$. This qualitative picture
shows the appearance of new peaks when considering the thermo-induced effect.
The explanation is based on the deformation of the wave functions. Indeed,
since the oscillation strength related to these peaks is proportional to the
squared modulus of the wave functions at zero electron-hole separation, only
the states associated with $\left|\psi_{nm}(\mathbf{r}=0)\right|^{2}\neq 0$
can be determined by the linear optical absorption spectrum. For zero-
temperature magnetoexciton, only the $s$-state wave functions have non-
vanished oscillation strength while they vanish for all the other states such
as $p$ and $d$; hence, we can only detect $s$-states exciton from the linear
optical response (Fig. 8 (a) and (b)). However, because of the thermal-induced
motional Stark effect, other states, such as $p$-states, now become the
superposition of other states, including $s$-states. Therefore, we get the
signal of the $p$-and-$d$-state-exciton peaks on the linear optical absorption
spectra at room temperature, as shown in Fig. 8(c).

##  IV Conclusion

In this work, by separating the c.m. motion from an exciton in monolayer
transition-metal dichalcogenide WSe2 with a magnetic field, we have pointed
out the motional Stark potential on the Schrödinger equation, which was
usually neglected. We proposed a new mechanism that the motional Stark
potential could affect the energy spectra of the magnetoexciton by the thermal
motion of its center of mass. As an observation from our calculation, the
thermal-induced motional Stark potential not only shifts the energy spectra
but also deforms the orbital of the magnetoexciton. The first consequence can
be directly observed by the shift of the resonance peaks on the absorption
spectra, while the second one indirectly changes the radiative lifetime of
free magnetoexciton. This change can be observed by examining the width of the
resonance peaks on the absorption spectra. Moreover, the thermal-induced
motional Stark effect also helps the $p-$state exciton peaks emerge on the
linear optical absorption spectra, which cannot happen at zero temperature.
These results potentially provide another aspect of studying the influence of
temperature on magnetoexciton. Finally, this mechanism could be extended for
not only neutral excitons but also trions, biexcitons in Dirac materials,
magnetoexcitons originated from the strain-induced pseudo magnetic field,
intervalley or interlayer excitons.

###### Acknowledgements.

D.-N. Ly and N.-H. P. are funded by Ho Chi Minh City University of Education
Foundation for Science and Technology under grant numbers CS.2019.19.43TD and
CS.2019.19.44TD. This work is funded by Foundation for Science and Technology
of Vietnam Ministry of Education and Training under grant number
B2022-SPS-09-VL. This work was carried out by the high-performance cluster at
Ho Chi Minh University of Education, Vietnam.

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this statement is that the unitarity of the Fourier transform and of the
Bargmann transform are equivalent. The results in this section show how
special the Gaussian function is, because the STFT with a Gaussian window is,
except for a weighting factor, an entire function. Hence, we can switch to
complex analysis to prove statements about $V_{g_{0}}$. This is the main
reason why much more results are known for the STFT with a Gaussian window
than with other windows. The fact that complex analysis in several variables
also differs greatly from the one-dimensional theory is the main reason why,
despite knowing much about STFTs with one-dimensional Gaussian windows, higher
dimensional Gaussian Gabor Systems still keep many mysteries.
Getting back to unitary representations of the Heisenberg Group, we may now
write the Heisenberg group as $\mathbf{H}=\mathbb{C}^{d}\times\mathbb{R}$. For
$z=x+i\omega\in\mathbb{C}^{d}$ and $w=\xi+i\eta$, we compute $B\rho(z,\tau)f$
for $f\in L^{2}(\mathbb{R}^{d})$. We start by writing $Bf(w)$ as a
representation coefficient. Recall that $\langle
f,\rho(x,\omega,\tau)g\rangle=e^{-2\pi i\tau}e^{\pi
ix\cdot\omega}V_{g}f(x,\omega)$. Then

$\displaystyle Bf(w)$ $\displaystyle=e^{-\pi
i\xi\cdot\eta}e^{\frac{\pi}{2}|w|^{2}}V_{g_{0}}f(\xi,-\eta)$ (10.49)
$\displaystyle=e^{-\pi i\xi\cdot\eta}e^{\frac{\pi}{2}|w|^{2}}e^{-\pi
i\xi\cdot(-\eta)}\langle f,\rho(\xi,-\eta,0)g_{0}\rangle$ (10.50)
$\displaystyle=e^{\frac{\pi}{2}|w|^{2}}\langle
f,\rho(\overline{w},0)g_{0}\rangle.$ (10.51)

Consequently,

$\displaystyle B(\rho(z,\tau)f)(w)$
$\displaystyle=e^{\frac{\pi}{2}|w|^{2}}\langle\rho(z,\tau)f,\rho(\overline{w},0)g_{0}\rangle$
(10.52) $\displaystyle=e^{\frac{\pi}{2}|w|^{2}}\langle
f,\rho(-z,-\tau)\rho(\overline{w},0)g_{0}\rangle$ (10.53)
$\displaystyle=e^{2\pi i\tau}e^{\frac{\pi}{2}|w|^{2}}e^{-\pi
i\Im(-z\cdot\overline{\overline{w}})}\langle
f,\rho(\overline{w}-z,0)g_{0}\rangle$ (10.54) $\displaystyle=e^{2\pi
i\tau}e^{\pi i\Im(w\cdot
z)}e^{\frac{\pi}{2}|w|^{2}}e^{-\frac{\pi}{2}|w-\overline{z}|^{2}}Bf(w-\overline{z})$
(10.55) $\displaystyle=e^{2\pi i\tau}e^{\pi w\cdot
z}e^{-\frac{\pi}{2}|z|^{2}}B(f(w-\overline{z})).$ (10.56)

Here, we used the fact that
$|w|^{2}-|w-\overline{z}|^{2}=2\Re(\overline{w}\cdot z)-|z|^{2}$ 262626Also,
note that the standard symplectic form is
$\sigma(z,w)=\sigma((x,\omega),(\xi,\eta))=\Im(z\cdot\overline{\omega})$..
Setting

$\beta(z,\tau)F(w)=e^{2\pi i\tau}e^{\pi w\cdot
z}e^{-\frac{\pi}{2}|z|^{2}}F(w-\overline{z}),$ (10.57)

we have derived the Bargmann representation of the Heisenberg group. Moreover,
we have shown that it is unitarily equivalent to the Schrödinger
representation via the Bargmann transform, i.e.,

$\beta(z,\tau)=\beta(x,\omega,\tau)=B\rho(x,\omega,\tau)B^{-1}.$ (10.58)

##  11 The Frame Set of a Window Function

One main issue in Gabor analysis is to determine all Gabor systems
$\mathcal{G}(g,\Gamma)$ which form a Gabor frame. This question, however, is
far too general to be answered completely. At the moment, the best we can hope
for is to determine all sets $\Gamma\subset\mathbb{R}^{2d}$ such that for a
fixed window $g$ the system $\mathcal{G}(g,\Gamma)$ is a Gabor frame. Even
this task is extremely challenging and no general method is known to solve
this problem. It is a case-by-case study and the tools to be used may differ
largely for different (classes of) windows. In order to have a more systematic
approach to this, still, very general problem, the notion of the frame set has
been introduced (see e.g., [21]).
However, we may distinguish between several types of frame sets and we are now
going to introduce the list which was set up in [9]. This generalizes the
ideas from [21].
For a fixed dimension $d$ and a fixed window $g$ we define:

  * •

The $\alpha$-frame set

$\mathfrak{F}_{\alpha}(g)=\\{\alpha\mathbb{Z}^{2d}\mid
G(g,\alpha\mathbb{Z}^{2d})\text{ is a frame}\\}$ (11.1)

  * •

The separable or $(\alpha,\beta)$-frame set

$\mathfrak{F}_{(\alpha,\beta)}(g)=\\{\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}\mid
G(g,\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d})\text{ is a frame}\\}$
(11.2)

For $d>1$, we allow $\alpha$ and $\beta$ to be multi-indices
($\alpha\mathbb{Z}^{d}=\alpha_{1}\mathbb{Z}\times\ldots\times\alpha_{d}\mathbb{Z}$,
same for $\beta$).

  * •

The lattice or $\Lambda$-frame set

$\mathfrak{F}_{\Lambda}(g)=\\{\Lambda\subset\mathbb{R}^{2d}\text{
lattice}\mid\mathcal{G}(g,\Lambda)\text{ is a frame}\\}$ (11.3)

  * •

The frame set

$\mathfrak{F}(g)=\\{\Gamma\subset\mathbb{R}^{2d}\mid\mathcal{G}(g,\Gamma)\text{
is a frame}\\}$ (11.4)

We note that we have the following chain of inclusions

$\mathfrak{F}_{\alpha}(g)\subset\mathfrak{F}_{(\alpha,\beta)}(g)\subset\mathfrak{F}_{\Lambda}(g)\subset\mathfrak{F}(g).$
(11.5)

We remark that in [9] also the symplectic frame set has been introduced

$\mathfrak{F}_{\sigma}(g)=\\{\Lambda_{\sigma}\subset\mathbb{R}^{2d}\text{
symplectic lattice}\mid\mathcal{G}(g,\Lambda_{\sigma})\text{ is a frame}\\}.$
(11.6)

The chain of inclusion given in [9] is however wrong. It is only true that

$\mathfrak{F}_{\alpha}(g)\subset\mathfrak{F}_{\sigma}(g)\subset\mathfrak{F}_{\Lambda}(g)\subset\mathfrak{F}(g),$
(11.7)

but, in general, $\Lambda_{(\alpha,\beta)}$ need not be in
$\mathfrak{F}_{\sigma}(g)$ even if $\mathcal{G}(g,\Lambda_{(\alpha,\beta)})$
is a frame. However, if $\alpha,\beta\in\mathbb{R}_{+}$ (and not multi-
indices), then
$\mathfrak{F}_{(\alpha,\beta)}(g)\subset\mathfrak{F}_{\sigma}(g)$.
It is of current interest to better understand Gabor systems with lattices
which are not symplectic. This is due to the fact that it is unknown how to
classify those lattices and how to build the proper metaplectic-like
operators. This problem does not occur in dimension $d=1$ as any 2-dimensional
lattice is symplectic.
In the case $d=1$, some frame sets are known, even though there has been
little progress until the work of Gröchenig and Stöckler [25]. The only frame
set of a (family of) function(s) where the entire frame set is known, is the
Gaussian, which is why we will treat it in a separate section. But before, we
will give a list of functions (ignoring the Gaussian) where at least the
separable frame set is known.

  * •

The one-sided exponential
$e^{-t}\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t)\;(\notin
M^{1}(\mathbb{R}))$ [33] (Janssen, 1996):

$\mathfrak{F}_{(\alpha,\beta)}(e^{-t}\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t))=\\{\alpha\mathbb{Z}\times\beta\mathbb{Z}\mid\alpha\beta\leq
1\\}$ (11.8)

  * •

The hyperbolic secant $\tfrac{1}{\cosh(\pi t)}\;(\in M^{1}(\mathbb{R}))$ [37]
(Janssen and Strohmer, 2002):

$\mathfrak{F}_{(\alpha,\beta)}(\cosh(\pi
t)^{-1})=\\{\alpha\mathbb{Z}\times\beta\mathbb{Z}\mid\alpha\beta<1\\}$ (11.9)

  * •

The two-sided exponential $e^{-|t|}\;(\in M^{1}(\mathbb{R}))$ [35] (Janssen,
2003):

$\mathfrak{F}_{(\alpha,\beta)}(e^{-|t|})=\\{\alpha\mathbb{Z}\times\beta\mathbb{Z}\mid\alpha\beta<1\\}$
(11.10)

We say that the above frame sets are full, as the necessary condition implied
by the corresponding Balian-Low theorem is also sufficient. We note that if
$\mathcal{G}(g,\alpha\mathbb{Z}\times\beta\mathbb{Z})$ is a frame, then so is
$\mathcal{G}(\mathcal{F}g,\beta\mathbb{Z}\times\alpha\mathbb{Z})$ by the
results on the symplectic and the metaplectic group in Section 7 272727Also
re-call the Fundamental Identity of Time-Frequency Analysis 2.5, which is
often interpreted in the way that the Fourier transform rotates the time-
frequency plane by $90^{\circ}$.. Therefore, the functions

  * •

$\mathcal{F}(e^{-t}\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t))(\omega)=\frac{1}{1+2\pi
i\omega}$

  * •

$\mathcal{F}(e^{-|t|})(\omega)=\frac{2}{1+4\pi^{2}\omega^{2}}=\frac{2}{(1+2\pi
i\omega)(1-2\pi i\omega)}$

possess full frame sets as well. The hyperbolic secant is a fixed-point of the
Fourier transform and, hence, its Fourier transform does not provide a new
window function with a full frame set. Gröchenig and Stöckler [25] realized
that the above mentioned window functions are so-called totally positive
functions of finite type.

######  Definition 11.1.

A non-constant, measurable function $g:\mathbb{R}\to\mathbb{R}$ is called
totally positive, if for any $N\in\mathbb{N}$ and any two sets of increasing
real numbers

$x_{1}<x_{2}<\ldots<x_{N}\qquad\textnormal{ and }\qquad
y_{1}<y_{2}<\ldots<y_{N},$ (11.11)

the matrix

$M_{g}=\left(g(x_{j}-y_{k})\right)_{j,k=1}^{N}$ (11.12)

is positive semidefinite, i.e.,

$x\cdot M_{g}x\geq 0,\qquad\forall x\in\mathbb{R}^{N}.$ (11.13)

There is a nice characterization of totally positive functions by means of
their Fourier transform due to I. Schoenberg [45, 46]282828Schoenberg’s
characterization was actually in terms of the two-sided Laplace transform, but
the characterization is basically the same., which might as well serve as
their definition.

######  Theorem 11.2 (Schoenberg).

A function $g:\mathbb{R}\to\mathbb{R}$ is totally positive if and only if its
Fourier transform can be written as

$\widehat{g}(\omega)=C\,e^{-\gamma\omega^{2}}e^{-2\pi
i\delta\omega}\prod_{k=1}^{\infty}\frac{e^{2\pi i\delta_{k}\omega}}{1+2\pi
i\delta_{k}\omega},$ (11.14)

with $C>0$, $\gamma\geq 0$, $\delta,\delta_{k}\in\mathbb{R}$ and
$0<\gamma+\sum_{k=1}^{\infty}\delta_{k}^{2}<\infty$.

As we are interested in Gabor systems with totally positive functions, we may
as well assume that $\delta=0$, as it corresponds to a time-shift of the
function, which has no effect on the properties of the Gabor system. Hence,
our functions of interest can be characterized by

$\widehat{g}(\omega)=C\,e^{-\gamma\omega^{2}}\prod_{k=1}^{\infty}\frac{e^{2\pi
i\delta_{k}\omega}}{1+2\pi i\delta_{k}\omega}.$ (11.15)

If $\gamma=0$ and only finitely many $\delta_{k}$ are different from zero,
then we call $g$ a totally positive function of finite type;

$\widehat{g}(\omega)=C\,\prod_{k=1}^{M}\frac{e^{2\pi
i\delta_{k}\omega}}{1+2\pi i\delta_{k}\omega},\qquad M\in\mathbb{N}.$ (11.16)

If $\gamma\neq 0$ and, still, only finitely many $\delta_{k}$ are different
from zero, we call $g$ a totally positive function of finite Gaussian type;

$\widehat{g}(\omega)=C\,e^{-\gamma\omega^{2}}\prod_{k=1}^{M}\frac{e^{2\pi
i\delta_{k}\omega}}{1+2\pi i\delta_{k}\omega},\qquad M\in\mathbb{N}.$ (11.17)

The following result is due to Gröchenig and Stöckler [25].

######  Theorem 11.3 (Gröchenig and Stöckler).

Let $g$ be a totally positive function of finite type $M\geq 2$, then

$\mathfrak{F}_{(\alpha,\beta)}(g)=\\{\alpha\mathbb{Z}\times\beta\mathbb{Z}\mid\alpha\beta<1\\}.$
(11.18)

We note that any totally positive function of finite type $M\geq 2$ is in the
modulation space $M^{1}(\mathbb{R})$ (i.e., in Feichtinger’s algebra
$S_{0}(\mathbb{R})$). Hence, the frame set of totally positive functions of
finite type is full.
Now, one might get the impression that the Balian-Low theorem is the only
restriction for separable Gabor systems. However, there is another function
for which the separable frame set is known, namely the characteristic function
of an interval. The first parts of the frame set where described by Janssen
and became known as Janssen’s tie [36]. The gaps where finally filled by Dai
and Sun [7]. Their work describes in 17 different cases and on more than 100
pages for which parameters the Gabor system with the characteristic function
of an interval and a separable lattice is a frame.

###  11.1 The Frame Set of a Gaussian

As we have seen, for some window classes, it is known that the necessary
condition imposed by the Balian-Low theorem is also sufficient for the Gabor
system $\mathcal{G}(g,\alpha\mathbb{Z}\times\beta\mathbb{Z})$ to be a frame.
If we ask for general point sets $\Gamma$, or even only lattices $\Lambda$,
then there is precisely one window (or actually one class of windows) for
which a full characterization of Gabor frames is known, namely the Gaussian
window $g_{0}$ (and its metaplectic deformations as well as scalar multiples).
In dimensions $d>1$, hardly anything is known about (general)
characterizations of Gabor frames.
We will exploit the properties of the Bargmann transform to formulate the
question on determining the frame set of a 1-dimensional (standard) Gaussian
as a sampling problem in the Bargmann-Fock space
$\mathcal{F}^{2}(\mathbb{C})$. We consider relatively separated (also called
uniformly discrete) sets $\Gamma\subset\mathbb{C}$, that is $\inf_{k\neq
l}|z_{k}-z_{l}|>0$ for all $z_{k},z_{l}\in\Gamma$. The set $\Gamma$ is called
a set of sampling for the Bargmann-Fock space $\mathcal{F}^{2}(\mathbb{C})$ if
there exist positive constants $0<A\leq B<\infty$ such that

$A\lVert
F\rVert_{\mathcal{F}^{2}}^{2}\leq\sum_{z\in\Gamma}|F(z)|^{2}e^{-\pi|z|^{2}}\leq
B\lVert F\rVert_{\mathcal{F}^{2}}^{2},\quad\forall
F\in\mathcal{F}^{2}(\mathbb{C}).$ (11.19)

The condition that $\Gamma\subset\mathbb{C}$ is relatively separated already
ensures that the upper frame bound is finite. Since the Bargmann transform $B$
maps $L^{2}(\mathbb{R})$ unitarily to $\mathcal{F}^{2}(\mathbb{C})$ by Theorem
10.4, we see that $\Gamma\subset\mathbb{C}$ is a set of sampling for
$\mathcal{F}^{2}(\mathbb{C})$ if and only if the real version of
$\Gamma\subset\mathbb{R}^{2}$ yields a Gabor frame for the standard Gaussian
$g_{0}$ 292929Again, we identify $\mathbb{C}$ and $\mathbb{R}^{2}$ in the
natural way; $\mathbb{C}\ni z=x+iy\,\leftrightarrow(x,y)\in\mathbb{R}^{2}$..

####  11.1.1 The Theorem of Lyubarskii and Seip and Wallstén

The following theorem is so important that it deserves to have an own section.
It still is one of the most celebrated results in Gabor analysis and was
proven independently by Lyubarskii [41] and Seip and Wallsten [47], [48] in
the year 1992.

######  Theorem 11.4.

  1. (i)

A discrete set $\Gamma\subset\mathbb{C}$ is a set of sampling for
$\mathcal{F}^{2}(\mathbb{C})$ if and only if $\Gamma$ can be expressed as a
finite union of relatively separated sets and if it contains a relatively
separated subset $\Gamma_{0}$ with $D^{-}(\Gamma_{0})>1$.

  2. (ii)

The Gabor system $\mathcal{G}(g_{0},\Gamma)$ is a frame for
$L^{2}(\mathbb{R})$ if and only if $\Gamma\subset\mathbb{R}^{2}$ is relatively
separated and its lower Beurling density satisfies $D^{-}(\Gamma)>1$.

We note that the two statements are equivalent by the Bargmann transform. The
proof relies, as pointed out above, on the Bargmann transform, which allows to
switch to complex analysis (in one variable) and the sampling problem in the
Bargmann-Fock space. In $\mathcal{F}^{2}(\mathbb{C})$, entire functions with
prescribed zeros can be constructed explicitly by means of (a version of) the
Weierstrass sigma function

$\sigma(z)=z\prod_{\gamma\in\Gamma\backslash
0}\left(1-\frac{z}{\gamma}\right)e^{\frac{z}{\gamma}+\frac{z^{2}}{2\gamma^{2}}}.$
(11.20)

The function $\tau(z)=\frac{\sigma(z)}{z}$ then takes the role of an
interpolating function 303030Note the similarity of $\tau(z)$ to the role of
sinc function $\frac{\sin(\pi x)}{\pi x}$ in the WNKS sampling theorem;
$\sin(\pi z)=\pi
z\prod_{k\in\mathbb{Z}/\\{0\\}}\left(1-\frac{z}{k}\right)e^{\frac{z}{k}},\quad
z\in\mathbb{C}.$ . Then, from the lower Beurling density of the zero set,
growth estimates for $\sigma$ are obtained. This leads to the result that
$\Gamma$ is a set of sampling if and only if its lower Beurling density is
(strictly) greater than 1.

###  11.2 The Frame Set in Higher Dimensions

As already mentioned, only few results on Gabor frames for
$L^{2}(\mathbb{R}^{d})$, $d>1$, are known. We state here the main result on
tensor products of frames from [3] (see also Appendix A.7).

######  Theorem 11.5.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two countable sets. The sequence
$(e_{\gamma_{k}})_{\gamma_{k}\in\Gamma_{k}}$ is a frame for the Hilbert space
$\mathcal{H}_{k}$, $k=1,2$, if and only if $(e_{\gamma_{1}}\otimes
e_{\gamma_{2}})_{(\gamma_{1},\gamma_{2})\in\Gamma_{1}\times\Gamma_{2}}$ is a
frame for $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$.

One consequence of the above theorem for higher dimensional Gabor systems is
that the necessary density condition can no longer be sufficient.

######  Example 11.6.

Consider the Hilbert space $L^{2}(\mathbb{R}^{2})\cong
L^{2}(\mathbb{R})\otimes L^{2}(\mathbb{R})$ and the corresponding standard
Gaussian313131We are a bit sloppy with our notation here as
$g_{0}(t)=2^{d/4}e^{-\pi|t|^{2}}$ denotes the standard Gaussian in any
dimension.

$g_{0}(t_{1},t_{2})=2^{1/2}e^{-\pi(t_{1}^{2}+t_{2}^{2})}=g_{0}(t_{1})\otimes
g_{0}(t_{2}).$ (11.21)

and consider the two Gabor systems $\mathcal{G}_{1}(g_{0},\mathbb{Z}^{2})$ and
$\mathcal{G}_{2}(g_{0},\alpha\mathbb{Z}^{2})$ with $\alpha<1$. Then, the Gabor
system $\mathcal{G}=\mathcal{G}_{1}\otimes\mathcal{G}_{2}$ is a (standard)
Gaussian Gabor system in $L^{2}(\mathbb{R}^{2})$ with arbitrarily high
density, but it can never be a Gabor frame for $L^{2}(\mathbb{R}^{2})$ because
$\mathcal{G}_{1}$ is not a Gabor frame for its respective Hilbert space
$L^{2}(\mathbb{R})$.

It is of course possible to construct a wealth of Gabor systems (not only
Gaussian) with arbitrarily high density which fail to be frames. Also, it is
straight forward to extend the method to tensor product Hilbert spaces of the
form $\mathcal{H}_{1}\otimes\ldots\otimes\mathcal{H}_{N}$.

##  Appendix A Appendices

###  A.1 More Uncertainty Principles

This is a collection of some more uncertainty principles without proof. The
interested reader is referred to [20] and [19].

######  Theorem A.1 (Lieb’s inequalities).

Assume $f,g\in L^{2}(\mathbb{R}^{d})$. Then

$\iint_{\mathbb{R}^{2d}}|V_{g}f(x,\omega)|^{p}\,d(x,\omega)\begin{cases}\leq\left(\tfrac{2}{p}\right)^{d}(\lVert
f\rVert_{2}\lVert g_{2}\rVert)^{p}&2\leq p<\infty\\\
\geq\left(\tfrac{2}{p}\right)^{d}(\lVert f\rVert_{2}\lVert
g_{2}\rVert)^{p}&1\leq p\leq 2\\\ \end{cases}$ (A.1)

These inequalities are quite deep results and their proof requires the sharp
constants in Hölder’s inequality. A consequence of Lieb’s inequalities is the
following result.

######  Theorem A.2 (Gröchenig).

Suppose that $\lVert f\rVert_{=}\lVert g\rVert_{2}=1$. If
$U\subset\mathbb{R}^{2d}$ and $\varepsilon\geq 0$ are chosen such that

$\iint_{U}|V_{g}f(x,\omega)|^{2}\,d(x,\omega)\geq(1-\varepsilon).$ (A.2)

Then, we have

$|U|\geq\left(\tfrac{p}{2}\right)^{\frac{2d}{p-2}}(1-\varepsilon)^{\frac{p}{p-2}},\qquad\forall
p>2.$ (A.3)

In particular,

$|U|\geq\sup_{p>2}\left(\tfrac{p}{2}\right)^{\frac{2d}{p-2}}(1-\varepsilon)^{\frac{p}{p-2}}\geq
2^{d}(1-\varepsilon)^{2},\qquad(p=4).$ (A.4)

The classical Heisenberg-Pauli-Weyl uncertainty principle is minimized by
dilated, time-frequency shifted Gauss functions. As the standard Gaussian is a
fixed point of the Fourier transform, one may therefore suspect that Gaussian
decay is the fastest possible simultaneous decay for a pair of Fourier
transforms $(f,\widehat{f})$. This guess is answered affirmatively by Hardy’s
Theorem.

######  Theorem A.3 (Hardy).

Let $f\in L^{2}(\mathbb{R}^{d})$ and assume that

$f(x)=\mathcal{O}(e^{-a\pi x^{2}})\quad\text{ and
}\quad\widehat{f}(\omega)=\mathcal{O}(e^{-b\pi\omega^{2}}),$ (A.5)

for some $a,b,>0$. Then, three cases occur;

  1. (i)

If $ab=1$, then $f(x)=ce^{-a\pi x^{2}}$.

  2. (ii)

If $ab>1$, then $f\equiv 0$.

  3. (iii)

If $ab<1$, then any finite linear combinations of Hermite functions satisfies
these decay conditions.

Once again, this uncertainty principle can be rephrased in terms of the
Rihaczek distribution.

Assume that $|Rf(x,\omega)|=\mathcal{O}(e^{-\pi(ax^{2}+b\omega)^{2}})$. If
$ab=1$, then $f(x)=Ce^{-a\pi x^{2}}$. If $ab>1$, then $f\equiv 0$.

According to Metatheorem C we may conjecture a Hardy-like uncertainty
principle for other time-frequency representations as well.

######  Theorem A.4 (Gröchenig, Zimmermann).

Assume that

$|V_{g}f(x,\omega)|=\mathcal{O}(e^{-\frac{\pi}{2}(ax^{2}+b\omega^{2})}),$
(A.6)

for some constants $a,b>0$. Then three cases can occur;

  1. (i)

If $ab=1$ and $V_{g}f\not\equiv 0$, then, both, $f$ and $g$ are scalar
multiples of a time-frequency shift of a Gaussian of the form $e^{-a\pi
x^{2}}$.

  2. (ii)

If $ab>1$, then either $f\equiv 0$ or $g\equiv 0$ (or both).

  3. (iii)

If $ab<1$, then the decay condition is satisfied whenever $f$ and $g$ are
finite linear combinations of Hermite functions.

###  A.2 Hilbert-Schmidt Operators

Let $\mathcal{H}$ be a separable Hilbert space. A bounded operator
$K:\mathcal{H}\to\mathcal{H}$ is called a Hilbert-Schmidt operator if

$\sum_{n\in\mathbb{N}}\lVert Ke_{n}\rVert_{\mathcal{H}}^{2}<\infty,$ (A.7)

for some orthonormal basis $\\{e_{n}\mid n\in\mathbb{N}\\}$ of $\mathcal{H}$.
The Hilbert-Schmidt norm of $K$ is given by

$\lVert K\rVert_{H.S.}=\left(\sum_{n\in\mathbb{N}}\lVert
Ke_{n}\rVert_{\mathcal{H}}^{2}\right)^{1/2}.$ (A.8)

This quantity is then independent of the choice of basis. The Hilbert-Schmidt
norm dominates the operator norm. Let $f=\sum_{n\in\mathbb{N}}c_{n}e_{n}$,
where $c_{n}=c_{n}(f)$, $\\{e_{n}\mid n\in\mathbb{N}\\}$ an orthonormal basis
and $\lVert f\rVert_{\mathcal{H}}=1=\sum_{n\in\mathbb{N}}|c_{n}|^{2}$, then

$\lVert K\rVert_{op}^{2}=\sup_{\stackrel{{\scriptstyle
f\in\mathcal{H}}}{{\lVert f\rVert_{\mathcal{H}}=1}}}\lVert
Kf\rVert_{\mathcal{H}}^{2}\leq\sum_{n\in\mathbb{N}}|c_{n}|^{2}\lVert
Ke_{n}\rVert_{\mathcal{H}}^{2}\leq\sum_{n\in\mathbb{N}}|c_{n}|^{2}\sum_{n\in\mathbb{N}}\lVert
Ke_{n}\rVert_{\mathcal{H}}^{2}=\lVert K\rVert_{H.S.}^{2}.$ (A.9)

If $\mathcal{H}=L^{2}(\mathbb{R}^{d})$ and $K$ is an integral operator with
integral kernel $k$, that is

$Kf(x)=\int_{\mathbb{R}^{d}}k(x,y)f(y)\,dy,$ (A.10)

then $K$ is Hilbert-Schmidt if and only if $k\in L^{2}(\mathbb{R}^{2d})$. In
this case

$\lVert K\rVert_{H.S.}=\lVert k\rVert_{2}.$ (A.11)

###  A.3 Neumann Series

If a linear operator $U$ on a Banach space $\mathcal{B}$ is close enough to
the identity operator $I$, then it is invertible. To be more precise, if
$\lVert U-I\rVert_{op}<1$, then U is invertible on $\mathcal{B}$ and the
inverse operator is given by the Neumann series

$U^{-1}=\sum_{k=0}^{\infty}(I-U)^{k}.$ (A.12)

Furthermore,

$\lVert U^{-1}\rVert_{op}\leq\frac{1}{1-\lVert I-U\rVert_{op}}$ (A.13)

The convergence of the series should be understood in the sense of the
operator norm

$\lVert U^{-1}-\sum_{k=1}^{n}(I-U)^{k}\rVert_{op}\to 0,\quad n\to\infty.$
(A.14)

For details we refer to [5, Thm. 2.2.3], [19, App. A.3] or [27, p. 48].

###  A.4 Toeplitz Matrices and Laurent Operators

For details on this section and further reading we refer to [4]. A Toeplitz
matrix $M$ is a (not necessarily square) matrix, which is constant on the
diagonals, i.e.,

$M=\begin{pmatrix}m_{0}&m_{-1}&m_{-2}&\ldots&\ldots&m_{-k}\\\
m_{1}&m_{0}&m_{-1}&m_{-2}&\ldots&m_{-(k-1)}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
m_{l}&\ldots&\ldots&\ldots&\ldots&\cdot\end{pmatrix}$ (A.15)

A Laurent operator is a bi-infinite (square) Toeplitz matrix acting on
$\ell^{2}(\mathbb{Z})$;

$L=\begin{pmatrix}\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\\
\ddots&m_{1}&m_{0}&m_{-1}&m_{-2}&m_{-3}&\ddots\\\
\ddots&m_{2}&m_{1}&m_{0}&m_{-1}&m_{-2}&\ddots\\\
\ddots&m_{3}&m_{2}&m_{1}&m_{0}&m_{-1}&\ddots\\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\end{pmatrix}$ (A.16)

The following theorem summaries the main results from [4, Chap. 1]

######  Theorem A.5.

Let $m(t)\in C(\mathbb{T})$ with Fourier expansion

$m(t)=\sum_{k\in\mathbb{Z}}m_{k}e^{2\pi ikt},$ (A.17)

and let $L(m)$ be the associated Laurent operator. Then

$\lVert L(m)\rVert_{op}=\lVert m\rVert_{\infty}\quad\text{ and }\quad\lVert
L(m)^{-1}\rVert_{op}\leq\lVert L(m^{-1})\rVert_{op}=\lVert
m^{-1}\rVert_{\infty}$ (A.18)

If $m$ is “nice” enough, then the bound for the inverse frame operator is
sharp as well.

###  A.5 Proof of Hudson’s Theorem

We will now prove that $W(f,f)=Wf>0$ if and only if $f$ is a Gaussian function
of the form

$f(t)=e^{-\pi t\cdot At+2\pi b\cdot t+c},$ (A.19)

where $A\in GL(\mathbb{C},d)$, $\Re(A)>0$ (i.e., positive definite) and
$A^{*}=\overline{A}^{T}=A$, $b\in\mathbb{C}^{d}$, $c\in\mathbb{C}$.

For the sufficiency, we note that any Gaussian function $f$ can be written as
a scalar multiple of a finite composition of metaplectic operators applied to
the standard Gaussian $g_{0}(t)=2^{d/4}e^{-\pi t^{2}}$ and an appropriate
time-frequency shift. By the symplectic covariance principle for the Wigner
transform, i.e., $W(f,g)(Sz)=W(\widehat{S}^{1}f,\widehat{S}^{-1}g)(z)$, $S\in
Sp(\mathbb{R},2d)$ and $\widehat{S}=\mu(S)$, it is therefore sufficient to
establish the sufficiency for the standard Gaussian. As

$Wg_{0}(x,\omega)=2^{d}e^{-2\pi(x^{2}+\omega^{2})},$ (A.20)

this part is done.

For the necessity part, we make use of the Bargmann transform and use some
complex analysis. Assume that $f\in L^{2}(\mathbb{R}^{d})$ (non-zero) and
$Wf\geq 0$. We take the inner product of $Wf$ with the Wigner distribution of
$g_{0}$ and note that, by using the covarince principle in Proposition 3.7,

$W(M_{-\omega}T_{x}g_{0})>0,\quad\forall(x,\omega)\in\mathbb{R}^{2d}.$ (A.21)

Therefore,

$\langle
Wf,W(M_{-\omega}T_{x}g_{0})\rangle=\iint_{\mathbb{R}^{2d}}Wf(\xi,\eta)W(M_{-\omega}T_{x}g_{0})(\xi,\eta)\,d(\xi,\eta)>0,\quad\forall(x,\omega)\in\mathbb{R}^{2d}.$
(A.22)

Now, we apply Moyal’s formula (3.65) and Proposition 10.2 for the Bargmann
transform to identify the inner product $\langle
Wf,W(M_{-\omega}T_{x}g_{0})\rangle$ as the Bargmann transform of $f$. Writing
$z=x+i\omega\in\mathbb{C}^{d}$, we obtain

$\displaystyle\langle Wf,W(M_{-\omega}T_{x}g_{0})\rangle$
$\displaystyle=|\langle f,M_{-\omega}T_{x}g_{0}\rangle|^{2}$ (A.23)
$\displaystyle=|V_{g_{0}}f(x,-\omega)|^{2}$ (A.24)
$\displaystyle=|Bf(z)|^{2}e^{-\pi|z|^{2}}.$ (A.25)

Since the entire function $Bf$ does not vanish by (A.22), there exists an
entire function $q(z)$, such that

$Bf(z)=e^{q(z)}.$ (A.26)

Furthermore, since

$|Bf(z)|e^{-\frac{\pi}{2}|z|^{2}}\leq\lVert
V_{g_{0}}f\rVert_{\infty}\leq\lVert f\rVert_{2}\lVert g_{0}\rVert_{2}=\lVert
f\rVert_{2},$ (A.27)

$Bf$ satisfies the growth estimate

$|Bf(z)|\leq\lVert f\rVert_{2}e^{\frac{\pi}{2}|z|^{2}}.$ (A.28)

By taking the logarithm, we obtain the estimate

$|\Re\left(q(z)\right)|\leq c+\frac{\pi}{2}|z|^{2}.$ (A.29)

It follows by Caratheodory’s inequality of complex analysis (see Appendix A.6)
that $q$ itself satisfies

$|q(z)|\leq C_{1}+C_{2}|z|^{2}.$ (A.30)

Therefore, $q$ must be a quadratic polynomial of the form

$q(z)=\pi A^{\prime}z^{2}+2\pi b^{\prime}\cdot z+c^{\prime}.$ (A.31)

The restriction of $Bf$ to vectors in $i\mathbb{R}^{d}$ is

$Bf(i\omega)e^{-\frac{\pi}{2}\omega^{2}}=e^{q(i\omega)-\frac{\pi}{2}\omega^{2}}=e^{-\pi(\frac{1}{2}+A^{\prime})\omega^{2}+2\pi
ib^{\prime}\cdot\omega+c^{\prime}},$ (A.32)

which is a Gaussian function. Next, we express the restriction of $V_{g_{0}}f$
to $\\{0\\}\times\mathbb{R}$ in two different ways. On one hand,

$V_{g_{0}}f(0,-\omega)=\langle
f,M_{-\omega}g_{0}\rangle=\mathcal{F}(f\,g_{0})(-\omega).$ (A.33)

On the other hand, by using Proposition 10.2 once more, we see that

$V_{g_{0}}f(0,-\omega)=Bf(i\omega)e^{-\frac{\pi}{2}\omega^{2}}.$ (A.34)

Therefore, $\mathcal{F}(f\,g_{0})$ is in $L^{2}(\mathbb{R}^{d})$ and is a
Gaussian function. It follows, by using the operator
$\mu(J)=(-i)^{d/2}\mathcal{F}$, that the function $f\,g_{0}$ is itself a
Gaussian function. This implies that $f$ itself is a Gaussian function and
since $f\in L^{2}(\mathbb{R}^{d})$, $\Re(A)>0$ is necessary.

$\square$

###  A.6 Caratheodory’s Inequality

This inequality provides size estimates for an analytic function $f$ when size
estimates are known only for the real part of $f$. As a special case we have
the following theorem for entire functions of one variable [2, p. 3].

######  Theorem A.6.

If $f$ is entire on $\mathbb{C}$ and $\Re f(z)\leq
A_{\varepsilon}|z|^{\alpha+\varepsilon}$ for all $z\in\mathbb{C}$, then $f$ is
a polynomial of degree at most $\alpha$.

######  Corollary A.7 (Caratheodory’s inequality).

If $f(z)=f(z_{1},\ldots,z_{d})$ is entire on $\mathbb{C}^{d}$ and

$\Re f(z)\leq D+E|z|^{2},$ (A.35)

then $f$ is a quadratic polynomial of the form $f(z)=Az^{2}+b\cdot z+c$ for
some $d\times d$ matrix over $\mathbb{C}$, $b\in\mathbb{C}^{d}$ and
$c\in\mathbb{C}$.

###  A.7 Tensor Hilbert Spaces

We repeat a few of the details on tensor Hilbert spaces given in [3]. Let
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be two Hilbert spaces with inner
product $\langle.,.\rangle_{k}$ and norm $\lVert.\rVert_{k}$, $k,=1,2$,
respectively. By $\mathcal{H}_{1}\otimes_{a}\mathcal{H}_{2}$ we denote the
algebraic tensor product of the two spaces. We can define an inner product on
$\mathcal{H}_{1}\otimes_{a}\mathcal{H}_{2}$ in the following way;

$\langle f_{1}\otimes f_{2},g_{1}\otimes g_{2}\rangle=\langle
f_{1},g_{1}\rangle_{1}\langle f_{2},g_{2}\rangle_{2},\quad\forall
f_{1},g_{1}\in\mathcal{H}_{1},\;\forall f_{2},g_{2}\in\mathcal{H}_{2}.$ (A.36)

The completion under this inner product is a Hilbert space, which we denote by
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. This is called the topological tensor
product of the Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$.

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