Towards verifications of Krylov complexity

Ryu Sasaki

Department of Physics and Astronomy, Tokyo University of Science, Noda
278-8510, Japan

###### Abstract

Krylov complexity is considered to provide a measure of the growth of
operators evolving under Hamiltonian dynamics. The main strategy is the
analysis of the structure of Krylov subspace
$\mathcal{K}_{M}(\mathcal{H},\eta)$ spanned by the multiple applications of
the Liouville operator $\mathcal{L}$ defined by the commutator in terms of a
Hamiltonian $\mathcal{H}$, $\mathcal{L}:=[\mathcal{H},\cdot]$ acting on an
operator $\eta$,
$\mathcal{K}_{M}(\mathcal{H},\eta)=\text{span}\\{\eta,\mathcal{L}\eta,\ldots,\mathcal{L}^{M-1}\eta\\}$.
For a given inner product $(\cdot,\cdot)$ of the operators, the orthonormal
basis $\\{\mathcal{O}_{n}\\}$ is constructed from
$\mathcal{O}_{0}=\eta/\sqrt{(\eta,\eta)}$ by Lanczos algorithm. The moments
$\mu_{m}=(\mathcal{O}_{0},\mathcal{L}^{m}\mathcal{O}_{0})$ are closely related
to the important data $\\{b_{n}\\}$ called Lanczos coefficients. I present the
exact and explicit expressions of the moments $\\{\mu_{m}\\}$ for 16 quantum
mechanical systems which are exactly solvable both in the Schrödinger and
Heisenberg pictures. The operator $\eta$ is the sinusoidal coordinate which is
the variable of the eigenpolynomials $\\{P_{n}(\eta)\\}$ of the exactly
solvable Hamiltonian $\mathcal{H}$.

##  1 Introduction

In the study of quantum chaos, Krylov complexity is proposed as a measure of
the growth of operators in the Heisenberg picture [1, 2, 3, 4, 5, 6]. The
definition of Krylov complexity is universal, that is, applicable to a very
complicated system as well as an extremely ordered system, for example, an
exactly solvable system in the Heisenberg picture. As materials to support the
validity of the concept of Krylov complexity by contrast [7, 8], I present one
of the most basic ingredients of the theory, the moments $\\{\mu_{m}\\}$
(2.20) of a special operator $\eta$ called sinusoidal coordinate of many
exactly solvable quantum systems in the Heisenberg picture. They are ten
discrete quantum mechanical systems of finite dimensions and two infinite ones
plus four ordinary one dimensional quantum mechanics. In all these systems,
the sinusoidal coordinate is the variable in the eigenpolynomials
$\\{P_{n}(\eta)\\}$ of the Hamiltonian $\mathcal{H}$. These polynomials all
belong to the Askey scheme of hypergeometric orthogonal polynomials [9, 10,
11].

This paper is organised as follows. In section 2, the basic concepts of Krylov
complexity are briefly recapitulated [1] through the orthonormalisation of
operators in a Krylov subspace [12] by Lanczos algorithm [13]. In section 3
the outline of exactly solvable discrete quantum systems in the Schrödinger
picture is briefly reviewed [14]. Corresponding solutions of Heisenberg
equation of motion [15, 16], which are the principal tool of the present
paper, are discussed in some detail in section 4. The main results of the
paper, the exact and explicit expressions of the moments $\\{\mu_{m}\\}$ are
derived for ten exactly solvable discrete quantum mechanics in section 5 and
four ordinary one-dimensional quantum mechanics in section 6. The inner
product including the Boltzmann factor is introduced in section 6 to deal with
unbounded Hamiltonians. The exact expressions of the moments for the bounded
and unbounded Hamiltonians look very similar as shown in two main Theorems 5.2
and 6.1. The final section is for a summary and some comments.

##  2 Orthonormalisation of operators in the Krylov subspace

Let us start with a brief review of the general setting of the Krylov
complexity along the line of the seminal work [1] and others [2, 3, 4, 5] in
order to introduce proper notions and appropriate notation. The
orthonormalisation of operators in the Krylov subspace is the basic ingredient
for the evaluation of Krylov complexity. For simplicity of presentation, the
discrete quantum mechanical systems are discussed at the beginning. That is,
the basic vector space is assumed to be $\mathbb{C}^{N+1}$, $N\in\mathbb{N}$
in this section and up to §5. Later in §6, infinite dimensional Hilbert space
is introduced for the treatment of the traditional quantum mechanical systems.

Let us begin with the notation. The ordinary Krylov subspace [12] is spanned
by a series of vectors generated by multiple applications of a certain
operator $\mathcal{H}$ on a vector $v$,

$K_{M}(\mathcal{H};v)\stackrel{{\scriptstyle\text{def}}}{{=}}\text{span}\\{v,\mathcal{H}v,\mathcal{H}^{2}v,\ldots,\mathcal{H}^{M-1}v\\}.$

For the evaluation of Krylov complexity [1, 3, 2, 4, 5], a different type of
subspaces is necessary. It is spanned by a series of operators generated by
multiple applications of adjoint actions of a Hamiltonian $\mathcal{H}$ on an
operator, to be named as $\eta$ throughout this paper,

$\mathcal{K}_{M}(\mathcal{H};\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\text{span}\\{\eta,\mathcal{L}\eta,\mathcal{L}^{2}\eta,\ldots,\mathcal{L}^{M-1}\eta\\},$
(2.1)

in which the Liouville operator $\mathcal{L}$ denotes the commutator by the
Hamiltonian $\mathcal{H}$;

$\mathcal{L}\eta\stackrel{{\scriptstyle\text{def}}}{{=}}[\mathcal{H},\eta],\quad\mathcal{L}^{2}\eta=[\mathcal{H},\mathcal{L}\eta],\quad\cdots.$
(2.2)

The Hamiltonian $\mathcal{H}$, $\eta$ and other operators $\mathcal{V}$,
$\mathcal{W}$ etc are all $(N+1)\times(N+1)$ matrices with the complex
components,

$\mathbb{C}\ni\mathcal{V}_{x\,y},\mathcal{W}_{x\,y},\qquad
x,y\in\mathcal{X}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{0,1,\ldots,N\\}.$
(2.3)

The Hamiltonian $\mathcal{H}$ is a positive semi-definite hermitian matrix,

$\mathcal{H}^{\dagger}=\mathcal{H},\quad\mathcal{H}^{*}_{x\,y}=\mathcal{H}_{y\,x},$
(2.4)

in which $\dagger$ denotes the hermitian conjugation and $*$ means the complex
conjugation. Throughout this paper the operator $\eta$ is assumed to be a real
diagonal matrix,

$\eta\stackrel{{\scriptstyle\text{def}}}{{=}}\text{diag}\\{\eta(0),\eta(1),\ldots,\eta(N)\\},\quad\eta(x)\in\mathbb{R},\
x\in\mathcal{X}.$ (2.5)

Two types of bra-ket notation are used. The bra $(\\!(x|$ and ket $|x)\\!)$
correspond to the $x$-th unit vector $\boldsymbol{e}_{x}$ in
$\mathbb{C}^{N+1}$, that is,

$(\\!(x|y)\\!)=\delta_{x\,y},\qquad(\\!(x|\mathcal{V}|y)\\!)=\mathcal{V}_{x\,y},\quad
x,y\in\mathcal{X}.$ (2.6)

The orthonormal basis corresponding to the eigenvectors of the Hamiltonian
$\mathcal{H}$ is denoted by the bra $\langle n|$ and the ket $|n\rangle$,

$\langle m|n\rangle=\delta_{m\,n},\quad\langle m|\mathcal{H}=\langle
m|\mathcal{E}(m),\quad\mathcal{H}|n\rangle=\mathcal{E}(n)|n\rangle,\quad\mathcal{E}(m),\mathcal{E}(n)\geq
0,\quad m,n\in\mathcal{X}.$ (2.7)

The inner product of operators $\mathcal{V}$ and $\mathcal{W}$ is defined by
the trace,

$(\mathcal{V},\mathcal{W})\stackrel{{\scriptstyle\text{def}}}{{=}}\text{Tr}[\\![\mathcal{V}^{\dagger}\mathcal{W}]\\!]=\sum_{x\in\mathcal{X}}(\\!(x|\mathcal{V}^{\dagger}\mathcal{W}|x)\\!)=\sum_{n\in\mathcal{X}}\langle
n|\mathcal{V}^{\dagger}\mathcal{W}|n\rangle=(\mathcal{W},\mathcal{V})^{*},$
(2.8)

which is real if $\mathcal{V}$ and $\mathcal{W}$ are both hermitian or anti-
hermitian. The norm of an operator $\mathcal{V}$ is denoted by,

$|\\!|\mathcal{V}|\\!|=\sqrt{(\mathcal{V},\mathcal{V})}=\sqrt{\text{Tr}[\\![\mathcal{V}^{\dagger}\mathcal{V}]\\!]}\geq
0.$ (2.9)

It should be stressed that the Liouville operator $\mathcal{L}$ flips from the
right to left side and vice versa, under the present definition of the inner
product,

$(\mathcal{V},\mathcal{L}\mathcal{W})=\text{Tr}[\\![\mathcal{V}^{\dagger}(\mathcal{H}\mathcal{W}-\mathcal{W}\mathcal{H})]\\!]=\text{Tr}[\\![(\mathcal{V}^{\dagger}\mathcal{H}-\mathcal{H}\mathcal{V}^{\dagger})\mathcal{W}]\\!]=\text{Tr}[\\![[\mathcal{H},\mathcal{V}]^{\dagger}\mathcal{W}]\\!]=(\mathcal{L}\mathcal{V},\mathcal{W}).$
(2.10)

It is also obvious that $(\mathcal{V},\mathcal{L}\mathcal{V})$ vanishes if
$\mathcal{V}$ is hermitian or anti-hermitian,

$\mathcal{V}^{\dagger}=\pm\mathcal{V}\Longrightarrow(\mathcal{V},\mathcal{L}\mathcal{V})=\pm\text{Tr}[\\![\mathcal{V}(\mathcal{H}\mathcal{V}-\mathcal{V}\mathcal{H})]\\!]=0.$
(2.11)

The orthonormalisation of the Krylov subspace
$\mathcal{K}_{m}(\mathcal{H},\eta)$ (2.1), $\\{\mathcal{O}_{n}\\}$,
$n=0,1,\ldots$ is much simpler than the ordinary Gram-Schmidt
orthonormalisation due to the built-in structure of
$\mathcal{K}_{m}(\mathcal{H},\eta)$. The orthonormalisation à la Lanczos [13]
starts with

$\displaystyle\mathcal{O}_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}\eta/|\\!|\eta|\\!|,\qquad|\\!|\eta|\\!|^{2}=\sum_{x\in\mathcal{X}}\eta(x)^{2},$
(2.12)
$\displaystyle\mathcal{W}_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{0},\quad
b_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{0}|\\!|=|\\!|\mathcal{L}\mathcal{O}_{0}|\\!|,\quad\mathcal{O}_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{1}}\mathcal{W}_{0}.$
(2.13)

By the above property (2.11), $\mathcal{O}_{1}$ is automatically orthogonal
with the previous $\mathcal{O}_{0}$,

$(\mathcal{O}_{j},\mathcal{O}_{l})=\delta_{j\,l},\quad j,l=0,1.$ (2.14)

The next step is the orthogonalisation of $\mathcal{L}\mathcal{O}_{1}$ with
$\mathcal{O}_{0}$,

$\mathcal{W}_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{1}-b_{1}\mathcal{O}_{0},\quad
b_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{1}|\\!|,\quad\mathcal{O}_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{2}}\mathcal{W}_{1}.$
(2.15)

The orthogonality $(\mathcal{O}_{1},\mathcal{W}_{1})=0$ is obvious by (2.11)
and (2.14). By construction, the orthogonality
$(\mathcal{O}_{0},\mathcal{W}_{1})=0$ holds. As

$\displaystyle(\mathcal{O}_{0},\mathcal{W}_{1})$
$\displaystyle=(\mathcal{O}_{0},\mathcal{L}\mathcal{O}_{1})-b_{1}(\mathcal{O}_{0},\mathcal{O}_{0})=(\mathcal{L}\mathcal{O}_{0},\mathcal{O}_{1})-b_{1}=\frac{1}{b_{1}}(\mathcal{L}\mathcal{O}_{0},\mathcal{L}\mathcal{O}_{0})-b_{1}=0,$

the orthonormality up to 2 is established

$(\mathcal{O}_{j},\mathcal{O}_{l})=\delta_{j\,l},\quad j,l=0,1,2.$ (2.16)

The flip property
$(\mathcal{V},\mathcal{L}\mathcal{W})=(\mathcal{L}\mathcal{V},\mathcal{W})$
(2.10) plays an important role in the above and further calculations. The
process goes on as

$\mathcal{W}_{k}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k-1},\quad
b_{k+1}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{k}|\\!|,\quad\mathcal{O}_{k+1}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{k+1}}\mathcal{W}_{k}.$
(2.17)

It is easy to prove that $\mathcal{O}_{k+1}$ is orthogonal to all the previous
ones $(\mathcal{O}_{j},\mathcal{O}_{k+1})=0$, $j=0,1,\ldots,k$, by assuming
the previous ones are orthonormal

$(\mathcal{O}_{j},\mathcal{O}_{l})=\delta_{j\,l},\quad j,l=0,1,\ldots,k.$
(2.18)

It is obvious by construction
$(\mathcal{O}_{k},\mathcal{W}_{k})=0=(\mathcal{O}_{k},\mathcal{O}_{k+1})$.
Likewise,

$\displaystyle(\mathcal{O}_{k-1},\mathcal{W}_{k})$
$\displaystyle=\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k}\bigr{)}=\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{O}_{k}\bigr{)}-b_{k}=\frac{1}{b_{k}}\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{W}_{k-1}\bigr{)}-b_{k}$
$\displaystyle=\frac{1}{b_{k}}\bigl{(}\mathcal{L}\mathcal{O}_{k-1},\mathcal{W}_{k-1}\bigr{)}-b_{k}=\frac{1}{b_{k}}\bigl{(}\mathcal{W}_{k-1}+b_{k-1}\mathcal{O}_{k-2},\mathcal{W}_{k-1}\bigr{)}-b_{k}=b_{k}-b_{k}=0,$

as
$(\mathcal{O}_{k-2},\mathcal{W}_{k-1})=b_{k}(\mathcal{O}_{k-2},\mathcal{O}_{k})=0$
by assumption. For $j\leq k-2$, the same logic goes

$\displaystyle(\mathcal{O}_{j},\mathcal{W}_{k})$
$\displaystyle=\bigl{(}\mathcal{O}_{j},\mathcal{L}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k-1}\bigr{)}=(\mathcal{O}_{j},\mathcal{L}\mathcal{O}_{k})=\frac{1}{b_{k}}(\mathcal{L}\mathcal{O}_{j},\mathcal{W}_{k-1})$
$\displaystyle=\frac{1}{b_{k}}(\mathcal{W}_{j}+b_{j}\mathcal{O}_{j-1},\mathcal{W}_{k-1})=0,$

and the induction is complete. $\square$

###### Remark 2.1

The orthonormalisation is complete when all the Lanczos coefficients
$\\{b_{n}\\}$ are determined. The orthonormalisation stops at
$\mathcal{O}_{k}$ when $b_{k+1}$ vanishes, $b_{k+1}=0$. Two explicit examples
of the stopped orthonormalisation will be shown in §5.1. Since the Hilbert
space is $\mathbb{C}^{N+1}$, the totality of the basis $\\{O_{n}\\}$ is less
than $(N+1)^{2}$.

###### Remark 2.2

The orthonormal basis $\mathcal{O}_{n}$ has the following structure

$\mathcal{O}_{n}=\sum_{j=0}^{[\tfrac{n}{2}]}c_{j}^{(n)}\mathcal{L}^{n-2j}\mathcal{O}_{0},\quad\bigl{(}i^{n}\mathcal{O}_{n}\bigr{)}^{\dagger}=i^{n}\mathcal{O}_{n},$
(2.19)

in which $[m]$ is the Gauss’s symbol meaning the greatest integer not
exceeding $m$. The squares of the Lanczos coefficients up to $n$,
$\\{b_{1}^{2},\ldots,b_{n}^{2}\\}$, which are the length squared of the basis
before normalisation, are expressed as rational functions of the moments
$\mu_{2m}$,

$\mu_{2m}\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{O}_{0},\mathcal{L}^{2m}\mathcal{O}_{0}),\quad
0\leq m\leq n,\quad\mu_{0}=1.$ (2.20)

For example (cf [17] Table 3-2),

$b_{1}^{2}=\mu_{2},\quad b_{2}^{2}=\frac{\mu_{4}}{\mu_{2}}-\mu_{2},\quad
b_{3}^{2}=\frac{\mu_{2}(\mu_{6}-2\mu_{2}\mu_{4}+\mu_{2}^{3})}{\mu_{2}(\mu_{4}-\mu_{2}^{2})}-\frac{\mu_{4}}{\mu_{2}}+\mu_{2}.$
(2.21)

Such formulas can be checked by considering the formal scaling properties,

$\mathcal{H}\to\lambda\mathcal{H}\
\Longrightarrow\mu_{2m}\to\lambda^{2m}\mu_{2m},\quad b_{n}\to\lambda b_{n}.$
(2.22)

###### Remark 2.3

In some work, a formula involving the determinant of Hankel matrix of moments

$b_{1}^{2}\cdots b_{n}^{2}={\rm det}(\mu_{i+j})_{0\leq i,j\leq n}$

was erroneously reported. The l.h.s. scales as $\lambda^{2n}$ and the diagonal
part of the r.h.s. matrix, $1\cdot\mu_{2}\cdots\mu_{2n}$, scales as
$\lambda^{n(n+1)}$.

###### Remark 2.4

In a monograph [17] Viswanath and Müller reported a recursive formula to
determine $\\{b_{1}^{2},\ldots,b_{K}^{2}\\}$ based on the knowledge of
$\mu_{2},\ldots,\mu_{2K}$. The formula was recapitulated in [1](A.5).

###### Definition 2.5

Krylov complexity $K(\mathcal{H},\eta;t)$ is defined [1, 2, 4, 5] based on the
Heisenberg operator solution of $\mathcal{O}(t)$ of $\eta$ and its projection
component $\varphi_{n}(t)$ on $\mathcal{O}_{n}$,

$K(\mathcal{H},\eta;t)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{n}n\varphi_{n}(t)^{2},\quad\varphi_{n}(t)\stackrel{{\scriptstyle\text{def}}}{{=}}\bigl{(}i^{n}\mathcal{O}_{n},\mathcal{O}(t)\bigr{)}\in\mathbb{R},\quad
n=1,2,\ldots,$ (2.23)

in which $\mathcal{O}(t)$ is the Heisenberg operator solution of $\eta$,

$\mathcal{O}(t)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{i\mathcal{H}t}\mathcal{O}_{0}e^{-i\mathcal{H}t}=e^{i\mathcal{H}t}\eta
e^{-i\mathcal{H}t}/|\\!|\eta|\\!|,\quad\bigl{(}\mathcal{O}(t),\mathcal{O}(t)\bigr{)}=1.$
(2.24)

Reflecting the unit norm of $\mathcal{O}(t)$, $\sum_{n}\varphi(t)^{2}=1$
holds.

In the rest of this paper I present the explicit forms of the Heisenberg
operator solution (2.24) and the moments $\\{\mu_{2m}\\}$ of more than dozen
quantum mechanical systems. Based on the exact knowledge of the moments, the
Lanczos coefficients $\\{b_{n}\\}$ and the functions $\\{\varphi_{n}(t)\\}$
(2.23) can be evaluated as precisely as wanted.

##  3 Exactly solvable discrete quantum mechanics

Here I present ten exactly solvable discrete quantum mechanical systems, see
[18] for a review. The eigenvectors of the Hamiltonians are the hypergeometric
orthogonal polynomials of the Askey scheme [9, 10, 11, 14]. They are the
Krawtchouk (K), Hahn (H), dual Han (dH), Racah (R), quantum $q$-Krawtchouk
(q$q$K), $q$-Krawtchouk ($q$K), affine $q$-Krawtchouk (a$q$K), $q$-Hahn
($q$H), dual $q$-Hahn (d$q$H) and $q$-Racah ($q$R) polynomials. The
Hamiltonian $\mathcal{H}$ of these exactly solvable quantum mechanics is a
tridiagonal $(N+1)\times(N+1)$ real symmetric matrix,

$\displaystyle\mathcal{H}_{x\,y}=\bigl{(}B(x)+D(x)\bigr{)}\delta_{x,y}\\!-\\!\sqrt{B(x)D(x+1)}\,\delta_{x+1,y}\\!-\\!\sqrt{B(x-1)D(x)}\,\delta_{x-1,y},\
x,y\in\mathcal{X},$ (3.1)
$\mathcal{H}=\left(\begin{array}[]{cccccc}\\!\\!B(0)&-\sqrt{B(0)D(1)}&0&\cdots&\cdots&0\\\
\\!\\!-\sqrt{B(0)D(1)}&B(1)+D(1)&-\sqrt{B(1)D(2)}&0&\cdots&\vdots\\\
\\!\\!0&-\sqrt{B(1)D(2)}&B(2)+D(2)&-\sqrt{B(2)D(3)}&\cdots&\vdots\\\
\\!\\!\vdots&\cdots&\cdots&\cdots&\cdots&\vdots\\\
\\!\\!\vdots&\cdots&\cdots&\cdots&\cdots&0\\\
\\!\\!0&\cdots&\cdots&-\sqrt{\\!B(N\\!\\!-\\!\\!2)D(N\\!\\!-\\!\\!1)}&B(N\\!\\!-\\!\\!1)\\!+\\!D(N\\!\\!-\\!\\!1)&-\sqrt{\\!B(N\\!\\!-\\!\\!1)D(N)}\\\
\\!\\!0&\cdots&\cdots&0&-\sqrt{B(N\\!-\\!1)D(N)}&D(N)\end{array}\right).$

in which the functions $B(x)$ and $D(x)$ are positive except for the boundary
conditions,

$B(x),D(x)>0,\quad D(0)=0,\quad B(N)=0.$ (3.2)

The orthonormal eigenvectors of the Hamiltonian $\mathcal{H}$ are

$\displaystyle\mathcal{H}|n\rangle=\mathcal{E}(n)|n\rangle,\quad\mathcal{E}(n)\geq
0,\quad n\in\mathcal{X},$ (3.3)
$\displaystyle(\\!(x|n\rangle=\phi_{0}(x)P_{n}(\eta)d_{n},\quad\sum_{x\in\mathcal{X}}\phi_{0}^{2}(x)P_{m}(\eta)P_{n}(\eta)=\frac{\delta_{m\,n}}{d_{n}^{2}}\Leftrightarrow\
\langle m|n\rangle=\delta_{m\,n},$ (3.4)

in which $P_{n}(\eta)$ is a degree $n$ polynomial in $\eta$. The sinusoidal
coordinate $\eta$ is a linear or quadratic function of $x$ or $q^{\pm x}$
($0<q<1$) which vanishes at $x=0$, $\eta(0)=0$ [14],

$\begin{split}&\text{(i)}\ \eta(x)=x,\ K,H,\quad\text{(ii)}\ \eta(x)=x(x+d),\
dH,R\\\ &\text{(iii)}\ \eta(x)=1-q^{x},\qquad\text{(iv)}\ \eta(x)=q^{-x}-1,\
qH,qqK,qK,aqK\\\ &\text{(v)}\ \eta(x)=(q^{-x}-1)(1-dq^{x}),\quad
dqH,qR\end{split}$ (3.5)

Likewise, the eigenvalue $\mathcal{E}$ is a linear or quadratic function of
$n$ or $q^{\pm n}$ which vanishes at $n=0$, $\mathcal{E}(0)=0$ [14],

$\begin{split}&\text{(i)}\ \mathcal{E}(n)=n,\ K,dH,\quad\text{(ii)}\
\mathcal{E}(n)=n(n+d),\ H,R\\\ &\text{(iii)}\ \mathcal{E}(n)=1-q^{n},\
qqK\qquad\text{(iv)}\ \mathcal{E}(n)=q^{-n}-1,\ dqH,aqK\\\ &\text{(v)}\
\mathcal{E}(n)=(q^{-n}-1)(1-dq^{n}),\quad qK,qH,qR.\end{split}$ (3.6)

In the formulas (3.5), (3.6) the parameter $d$ is specific in each system. As
functions of $x$, $\\{P_{n}\bigl{(}\eta(x)\bigr{)}\\}$ are terminating
($q$)-hypergeometric functions [9, 10, 11] and they are normalised by a
uniform condition,

$P_{n}(0)=1,\quad n\in\mathcal{X}.$ (3.7)

As orthogonal polynomials in $\eta$ with the above normalisation condition,
$\\{P_{n}(\eta)\\}$ satisfy three term recurrence relation [14],

$\displaystyle\eta P_{n}(\eta)$
$\displaystyle=A_{n}\bigl{(}P_{n+1}(\eta)-P_{n}(\eta)\bigr{)}+C_{n}\bigl{(}P_{n-1}(\eta)-P_{n}(\eta)\bigr{)},$
(3.8)

in which the coefficients $A_{n}$, $C_{n}$ are negative except for the
boundary conditions

$A_{n},\ C_{n}<0,\quad C_{0}=0,\quad A_{N}=0,$ (3.9)

so that $P_{-1}$ and $P_{N+1}$ do not enter into the theory.

For definiteness, I show the data of the simplest example, the Krawtchouk (K)
system:

$\displaystyle B(x)=p(N-x),\quad D(x)=(1-p)x,\qquad 0<p<1,$ $\displaystyle
P_{n}(\eta)=P_{n}(x)={}_{2}F_{1}\Bigl{(}\genfrac{}{}{0.0pt}{}{-n,\,-x}{-N}\Bigm{|}p^{-1}\Bigr{)},\quad\mathcal{E}(n)=n,\quad\eta(x)=x,$
$\displaystyle\phi_{0}^{2}(x)=\frac{N!}{x!\,(N-x)!}\Bigl{(}\frac{p}{1-p}\Bigr{)}^{x},\quad
d_{n}^{2}=\frac{N!}{n!\,(N-n)!}\Bigl{(}\frac{p}{1-p}\Bigr{)}^{n}\times(1-p)^{N},$
$\displaystyle A_{n}=-p(N-n),\quad C_{n}=-(1-p)n.$

As will be shown shortly, most of the data, except for $\eta$, $\mathcal{E}$,
$A_{n}$ and $C_{n}$, are not needed for the evaluation of the moments
$\\{\mu_{2m}\\}$ and the functions $\\{\varphi_{n}(t)\\}$.

##  4 Solutions of Heisenberg equation of motion

As shown in (2.24), the explicit form of the Heisenberg operator solution is
essential for the determination of the functions $\\{\varphi_{n}(t)\\}$.
Although the Heisenberg solution of the harmonic oscillator potential
($x^{2}$) was known in the early days of quantum mechanics, it was late 1970’s
that those for four other potentials in one-dimensional quantum mechanics were
reported by Nieto and Simmons [19]. They were for the potentials
$x^{2}+1/x^{2}$, $1/\sin^{2}x$, $-1/\cosh^{2}x$ and the Morse potential. The
term ‘sinusoidal coordinate’ was also coined by them, meaning that $\eta$
undergoes sinusoidal motion with frequencies depending on the energy. About a
quarter century after Nieto and Simmons, the list of exact Heisenberg operator
solutions was enlarged by Odake and myself to include many discrete quantum
mechanics [14, 15, 16, 20] and some multi-particle dynamics [21].

The essence is the discovery [16] (this paper will be cites as I) that the
Hamiltonian $\mathcal{H}$ and the sinusoidal coordinate $\eta$ of exactly
sovable systems in the Schrödinger picture all satisfy a simple commutation
relation

$[\mathcal{H},[\mathcal{H},\eta]\,]=\eta\,R_{0}(\mathcal{H})+[\mathcal{H},\eta]\,R_{1}(\mathcal{H})+R_{-1}(\mathcal{H}).$
(4.1)

in which $R_{0}(\mathcal{H})$, $R_{1}(\mathcal{H})$ and $R_{-1}(\mathcal{H})$
are polynomials in $\mathcal{H}$ of maximal degree 2, 1 and 2, respectively,
reflecting the power counting of $\mathcal{H}$ on both sides. Those
$R_{i}(\mathcal{H})$’s may contain some system parameters but not dynamical
operators other than $\mathcal{H}$. In terms of the Liouville operator
$\mathcal{L}$, the obove commutation relation reads

$\mathcal{L}^{2}\eta=\eta\,R_{0}(\mathcal{H})+\mathcal{L}\eta\,R_{1}(\mathcal{H})+R_{-1}(\mathcal{H}).$
(4.2)

which is obviously generalised to

$\mathcal{L}^{m}\eta=\eta\,\mathcal{A}_{m}(\mathcal{H})+\mathcal{L}\eta\,\mathcal{B}_{m}(\mathcal{H})+\mathcal{C}_{m}(\mathcal{H}),\quad
m\in\mathbb{N}_{\geq 0},$

with the obvious initial conditions

$\begin{split}&\mathcal{A}_{0}(\mathcal{H})=1,\quad\mathcal{A}_{1}(\mathcal{H})=0,\quad\mathcal{A}_{2}(\mathcal{H})=R_{0}(\mathcal{H}),\\\
&\mathcal{B}_{0}(\mathcal{H})=0,\quad\
\mathcal{B}_{1}(\mathcal{H})=1,\quad\mathcal{B}_{2}(\mathcal{H})=R_{1}(\mathcal{H}),\\\
&\mathcal{C}_{0}(\mathcal{H})=0,\quad\ \,\mathcal{C}_{1}(\mathcal{H})=0,\quad\
\mathcal{C}_{2}(\mathcal{H})=R_{-1}(\mathcal{H}).\end{split}$ (4.3)

By solving the recurrence relations,

$\displaystyle\mathcal{A}_{m+1}(\mathcal{H})=R_{0}(\mathcal{H})\mathcal{B}_{m}(\mathcal{H}),\qquad\mathcal{B}_{m+1}(\mathcal{H})=\mathcal{A}_{m}(\mathcal{H})+R_{1}(\mathcal{H})\mathcal{B}_{m}(\mathcal{H}),$
$\displaystyle\mathcal{C}_{m+1}(\mathcal{H})=R_{-1}(\mathcal{H})\mathcal{B}_{m}(\mathcal{H})=\frac{R_{-1}(\mathcal{H})}{R_{0}(\mathcal{H})}\mathcal{A}_{m+1}(\mathcal{H}),$

one arrives at,

$\displaystyle\mathcal{L}^{m}\eta=$
$\displaystyle\,\eta\,\mathcal{A}_{m}(\mathcal{H})+\mathcal{L}\eta\,\mathcal{B}_{m}(\mathcal{H})+\mathcal{C}_{m}(\mathcal{H}),\qquad
m\in\mathbb{N}_{\geq 0},$ (4.4)
$\displaystyle\mathcal{A}_{m}(\mathcal{H})=R_{0}(\mathcal{H})\frac{\alpha_{+}(\mathcal{H})^{m-1}-\alpha_{-}(\mathcal{H})^{m-1}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})},$
(4.5)
$\displaystyle\mathcal{B}_{m}(\mathcal{H})=\frac{\alpha_{+}(\mathcal{H})^{m}-\alpha_{-}(\mathcal{H})^{m}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})},$
(4.6)
$\displaystyle\mathcal{C}_{m}(\mathcal{H})=R_{-1}(\mathcal{H})\frac{\alpha_{+}(\mathcal{H})^{m-1}-\alpha_{-}(\mathcal{H})^{m-1}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})},\quad
m\geq 1,$ (4.7)

in which two operators $\alpha_{\pm}(\mathcal{H})$ are the roots of the
quadratic equation,

$\displaystyle\alpha(\mathcal{H})^{2}-R_{1}(\mathcal{H})\alpha(\mathcal{H})-R_{0}(\mathcal{H})=0,$
(4.8)
$\displaystyle\alpha_{\pm}(\mathcal{H})=\frac{1}{2}\left(R_{1}(\mathcal{H})\pm\sqrt{R_{1}(\mathcal{H})^{2}+4R_{0}(\mathcal{H})}\right),$
(4.9)
$\displaystyle\alpha_{+}(\mathcal{H})+\alpha_{-}(\mathcal{H})=R_{1}(\mathcal{H}),\quad\alpha_{+}(\mathcal{H})\alpha_{-}(\mathcal{H})=-R_{0}(\mathcal{H}).$
(4.10)

They also satisfy the relations as shown in [16](I.2.22) and (I.2.23)

$\displaystyle\mathcal{E}(n+1)-\mathcal{E}(n)=\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)},\quad\mathcal{E}(n-1)-\mathcal{E}(n)=\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}.$
(4.11)
$\displaystyle\mathcal{E}(n)-\mathcal{E}(n-1)=\alpha_{+}\bigl{(}\mathcal{E}(n-1)\bigr{)},\quad\mathcal{E}(n)-\mathcal{E}(n+1)=\alpha_{-}\bigl{(}\mathcal{E}(n+1)\bigr{)},$
(4.12)
$\displaystyle\alpha_{+}\bigl{(}\mathcal{E}(n-1)\bigr{)}=-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)},\quad\alpha_{-}\bigl{(}\mathcal{E}(n+1)\bigr{)}=-\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}.$
(4.13)

Summing up $\\{\mathcal{L}^{m}\eta\\}$ leads to the exact Heisenberg operator
solution of $\eta$ [15, 16],

$\displaystyle e^{it\mathcal{H}}\eta\,e^{-it\mathcal{H}}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(it)^{m}}{m!}\mathcal{L}^{m}\eta,$
$\displaystyle=\mathcal{L}\eta\cdot\frac{e^{i\alpha_{+}(\mathcal{H})t}-e^{i\alpha_{-}(\mathcal{H})t}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})}-R_{-1}(\mathcal{H})/R_{0}(\mathcal{H})$
$\displaystyle\quad+\bigl{(}\eta+R_{-1}(\mathcal{H})/R_{0}(\mathcal{H})\bigr{)}\frac{-\alpha_{-}(\mathcal{H})e^{i\alpha_{+}(\mathcal{H})t}+\alpha_{+}(\mathcal{H})e^{i\alpha_{-}(\mathcal{H})t}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})}.$
(4.14)

Strictly speaking, $\eta$ on both sides represents the initial value at $t=0$
of the time-dependent Heisenberg operator. It should be stressed that the
derivation from (4.1) to (4.4)–(4.7) and (4.14) is purely algebraic. It is
valid in any quantum system, not necessarily finite dimensional ones. As shown
above, $\eta$’s time development is a superposition of two sinusoidal waves of
frequencies $\alpha_{\pm}\bigl{(}\mathcal{H}\bigr{)}$ depending on the total
energy.

A simple and useful information is extracted from the above expression (4.14),
the energy level dependence of the expectation value of the Heisenberg
operator $e^{it\mathcal{H}}\eta\,e^{-it\mathcal{H}}$, which is obviously time-
independent $\langle
n|e^{it\mathcal{H}}\eta\,e^{-it\mathcal{H}}|n\rangle=\langle n|\eta|n\rangle$.
Sandwiching (4.14) by $\langle n|$ and $|n\rangle$ and noting $\langle
n|\mathcal{L}\eta|n\rangle=0$, one obtains

$\displaystyle\left(\langle
n|\eta|n\rangle+\frac{R_{-1}\bigl{(}\mathcal{E}(n)\bigr{)}}{R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}}\right)\times\left(1-\frac{-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}e^{i\alpha_{+}\left(\mathcal{E}(n)\right)t}+\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}e^{i\alpha_{-}\left(\mathcal{E}(n)\right)t}}{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}}\right)=0,$
$\displaystyle\hskip 113.81102pt\Longrightarrow\langle
n|\eta|n\rangle=-\frac{R_{-1}\bigl{(}\mathcal{E}(n)\bigr{)}}{R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}},\qquad
n\in\mathcal{X}.$ (4.15)

as the second factor is time-dependent and non-vanishing.

##  5 Evaluation of $\mu_{2m}$ of exactly solvable discrete quantum mechanics

With the formulas (4.4)-(4.7) and (4.15), the derivation of the exact
expressions of the moment $\mu_{m}$ (2.20) is straightforward. For the
evaluation of moment ($m\geq 1$),

$\displaystyle\mu_{m}$ $\displaystyle=\sum_{n\in\mathcal{X}}\langle
n|\mathcal{O}_{0}\mathcal{L}^{m}\mathcal{O}_{0}|n\rangle=\sum_{n\in\mathcal{X}}\langle
n|\eta\mathcal{L}^{m}\eta|n\rangle/|\\!|\eta|\\!|^{2}$
$\displaystyle=\sum_{n\in\mathcal{X}}\left\\{\langle
n|\eta^{2}|n\rangle\mathcal{A}_{m}(\mathcal{H})|n\rangle+\langle
n|\eta\mathcal{L}\eta|n\rangle\mathcal{B}_{m}(\mathcal{H})|n\rangle+\langle
n|\eta|n\rangle\mathcal{C}_{m}(\mathcal{H})|n\rangle\right\\}/|\\!|\eta|\\!|^{2},$
(5.1)

all the coefficient terms $\mathcal{A}_{m}(\mathcal{H})|n\rangle$,
$\mathcal{B}_{m}(\mathcal{H})|n\rangle$,
$\mathcal{C}_{m}(\mathcal{H})|n\rangle$ are known

$\mathcal{H}|n\rangle=\mathcal{E}(n)|n\rangle\Rightarrow\
\alpha_{\pm}(\mathcal{H})|n\rangle=\alpha_{\pm}\bigl{(}\mathcal{E}(n)\bigr{)}|n\rangle,\quad
R_{i}(\mathcal{H})|n\rangle=R_{i}\bigl{(}\mathcal{E}(n)\bigr{)}|n\rangle,\quad
i=-1,0,1.$

One only needs the values of

$\langle n|\eta^{2}|n\rangle,\quad\langle
n|\eta\mathcal{L}\eta|n\rangle,\quad\langle n|\eta|n\rangle,$

and they are expressed by the coefficients $A_{n}$ and $C_{n}$ of the three
term recurrence relation (3.8). By using the explicit expression
$(\\!(x|n\rangle=\phi_{0}(x)P_{n}(\eta)d_{n}$ and the three term recurrence
relation, one obtains

$\displaystyle\eta|n\rangle=-(A_{n}+C_{n})|n\rangle+A_{n}\frac{d_{n}}{d_{n+1}}|n+1\rangle+C_{n}\frac{d_{n}}{d_{n-1}}|n-1\rangle,$
$\displaystyle\langle
n|\eta|n\rangle=-(A_{n}+C_{n})=-\frac{R_{-1}\bigl{(}\mathcal{E}(n)\bigr{)}}{R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}},$
(5.2) $\displaystyle\langle n+1|\eta|n\rangle=\langle
n|\eta|n+1\rangle=A_{n}\frac{d_{n}}{d_{n+1}},\quad\langle
n-1|\eta|n\rangle=\langle n|\eta|n-1\rangle=C_{n}\frac{d_{n}}{d_{n-1}}.$ (5.3)
$\displaystyle\eta|n+1\rangle=-(A_{n+1}+C_{n+1})|n+1\rangle+A_{n+1}\frac{d_{n+1}}{d_{n+2}}|n+2\rangle+C_{n+1}\frac{d_{n+1}}{d_{n}}|n\rangle,$
$\displaystyle\langle n|\eta|n+1\rangle=C_{n+1}\frac{d_{n+1}}{d_{n}},$ (5.4)
$\displaystyle\eta|n-1\rangle=-(A_{n-1}+C_{n-1})|n-1\rangle+A_{n-1}\frac{d_{n-1}}{d_{n}}|n\rangle+C_{n-1}\frac{d_{n-1}}{d_{n-2}}|n-2\rangle,$
$\displaystyle\langle n|\eta|n-1\rangle=A_{n-1}\frac{d_{n-1}}{d_{n}},$ (5.5)
$\displaystyle\Longrightarrow\quad\langle
n|\eta|n+1\rangle^{2}=A_{n}C_{n+1},\quad\langle
n|\eta|n-1\rangle^{2}=A_{n-1}C_{n},\quad n\geq 1.$ (5.6)

By combining them, one arrives at

$\displaystyle\langle n|\eta^{2}|n\rangle$ $\displaystyle=\sum_{\ell=n,n\pm
1}\langle n|\eta|\ell\rangle\langle\ell|\eta|n\rangle=\langle
n|\eta|n\rangle^{2}+\langle n|\eta|n+1\rangle^{2}+\langle
n|\eta|n-1\rangle^{2}$
$\displaystyle=(A_{n}+C_{n})^{2}+A_{n}C_{n+1}+A_{n-1}C_{n},$ (5.7)
$\displaystyle\langle n|\eta\mathcal{L}\eta|n\rangle$ $\displaystyle=\langle
n|\eta\mathcal{H}\eta|n\rangle-\langle n|\eta^{2}\mathcal{H}|n\rangle$
$\displaystyle=\sum_{\ell=n,n\pm 1}\langle
n|\eta|\ell\rangle\mathcal{E}(\ell)\langle\ell|\eta|n\rangle-\mathcal{E}(n)\langle
n|\eta^{2}|n\rangle$
$\displaystyle=(A_{n}+C_{n})^{2}\mathcal{E}(n)+A_{n}C_{n+1}\mathcal{E}(n+1)+A_{n-1}C_{n}\mathcal{E}(n-1)$
$\displaystyle\quad-\mathcal{E}(n)\Bigl{(}(A_{n}+C_{n})^{2}+A_{n}C_{n+1}+A_{n-1}C_{n}\Bigr{)}$
$\displaystyle=A_{n}C_{n+1}\,\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}+A_{n-1}C_{n}\,\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)},$
(5.8)

by using (4.11). It is easy to see that
$(A_{n}+C_{n})^{2}\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$ cancels
$\langle n|\eta|n\rangle\mathcal{C}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$ in
(5.1) due to (4.15). These lead to

$\displaystyle|\\!|\eta|\\!|^{2}\mu_{m}$ $\displaystyle=\
\sum_{n\in\mathcal{X}}A_{n}C_{n+1}\Bigl{(}A_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}B_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\Bigr{)}$
$\displaystyle\ \
+\sum_{n\in\mathcal{X}}A_{n-1}C_{n}\Bigl{(}A_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}B_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\Bigr{)}.$

With
$R_{0}\bigl{(}(\mathcal{E}(n)\bigr{)}=-\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}$
(4.10),

$\displaystyle\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$
$\displaystyle=-\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\frac{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m-1}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m-1}}{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}}$
$\displaystyle\qquad+\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\frac{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}}{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}}=\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m},$
(5.9)

and

$\displaystyle\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$
$\displaystyle=-\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\frac{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m-1}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m-1}}{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}}$
$\displaystyle\qquad+\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\frac{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}}{\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}-\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}}=\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m},$
(5.10)

one arrives at

$\displaystyle|\\!|\eta|\\!|^{2}\mu_{m}$
$\displaystyle=\sum_{n=0}^{N-1}A_{n}C_{n+1}\Bigl{(}\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\Bigr{)}^{m}+\sum_{n=1}^{N}A_{n-1}C_{n}\Bigl{(}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\Bigr{)}^{m},$

in which the boundary conditions (3.9) $A_{N}=0$ ans $C_{0}=0$ are used.
Changing $n-1\to n$ in the second sum leads to

$\displaystyle|\\!|\eta|\\!|^{2}\mu_{m}$
$\displaystyle=\sum_{n=0}^{N-1}A_{n}C_{n+1}\Bigl{(}\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}+\alpha_{-}\bigl{(}\mathcal{E}(n+1)\bigr{)}^{m}\bigr{)}$
$\displaystyle=\sum_{n=0}^{N-1}A_{n}C_{n+1}\left(\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}+\Bigl{(}-\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\Bigr{)}^{m}\right)$
$\displaystyle=2\sum_{n=0}^{N-1}A_{n}C_{n+1}\left(\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\right)^{m}\times\left\\{\begin{array}[]{cc}1&m:\
\text{even}\\\ 0&m:\ \text{odd}\end{array}\right.,$

in which (4.13) is used.

###### Remark 5.1

It should be stressed that the product of the coefficients of the three term
recurrence relations (3.8) $A_{n}C_{n+1}$ ( $A_{n-1}C_{n}$) is independent of
the normalisation change of the polynomial $P_{n}(\eta)\to
P_{n}(\eta)^{\prime}=\gamma_{n}P_{n}(\eta)$, $n\in\mathbb{Z}_{\geq 0}$.
$\gamma_{n}\neq 0$.

These results are summarised as the following

###### Theorem 5.2

The moments of the exactly solvable discrete quantum systems have a very
simple exact expression

$\mu_{2m}=2\sum_{n=0}^{N-1}A_{n}C_{n+1}\left(\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\right)^{2m}/|\\!|\eta|\\!|^{2},\quad\mu_{2m-1}=0,\quad
m\in\mathbb{N}.$ (5.11)

It applies to ten systems related to the Krawtchouk (K), Hahn (H), dual Han
(dH), Racah (R), quantum $q$-Krawtchouk (q$q$K), $q$-Krawtchouk ($q$K), affine
$q$-Krawtchouk (a$q$K), $q$-Hahn ($q$H), dual $q$-Hahn (d$q$H) and $q$-Racah
($q$R) polynomials.

In the rest of this section, the necessary data for the evaluation of the
moments $\mu_{2m}$ of the ten discrete quantum systems are provided. They are
$B(x)$ and $D(x)$ for the definition of the Hamiltonian $\mathcal{H}$ and for
specifying the parameter ranges, the sinusoidal coordinate $\eta(x)$, the
energy eigenvalues $\mathcal{E}(n)$, $R_{0}(\mathcal{H})$ and
$R_{1}(\mathcal{H})$ for the derivation of
$\alpha_{\pm}\bigl{(}\mathcal{E}(n)\bigr{)}$ and the coefficients of the three
term recurrence relation $A_{n}$ and $C_{n}$. It should be stressed that for
these exactly solvable systems, the inside of the square root
$R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}$
(4.9) is always a complete square and
$\alpha_{\pm}\bigl{(}\mathcal{E}(n)\bigr{)}$ are polynomials in $n$ or $q^{\pm
n}$ of maximal degree two. For the full details of these exactly solvable
discrete quantum systems, a paper by Odake and myself [14] should be
consulted.

###  5.1 Krawtchouk and dual Hahn

These two systems have $\mathcal{E}(n)=n$ and $R_{0}=1$ and $R_{1}=0$ so that
$\alpha_{\pm}\bigl{(}\mathcal{E}(n)\bigr{)}=\pm 1$ and all the moments are
identical

$\text{K},\text{dH}:\qquad\mu_{2m}=\mu_{2}=2\sum_{n=0}^{N-1}A_{n}C_{n+1}/|\\!|\eta|\\!|^{2},\quad
m\in\mathbb{N}.$ (5.12)

This means by (2.21)

$b_{1}^{2}=\mu_{2},\quad b_{2}^{2}=1-\mu_{2},\quad b_{3}^{2}=0.$ (5.13)

These two examples shed light on an important property of the moments as
stated in the following

###### Proposition 5.3

When all the moments are equal, $\mu_{2m}=\mu_{2}$, $m\in\mathbb{N}$, the
Lanczos orthogonalisation stops at $\mathcal{O}_{2}$. Likewise, when all the
moments form a geometrical sequence $\mu_{2m}=\lambda^{2(m-1)}\mu_{2}$,
$m\in\mathbb{N}$, $\lambda\in\mathbb{R}_{>0}$, the Lanczos orthogonalisation
stops at $\mathcal{O}_{2}$. This case reduces to the constant case by the
scaling of the Hamiltonian $\mathcal{H}\to\lambda^{-1}\mathcal{H}$, which is
absorbed by the time rescaling $t\to\lambda t$. The very early stopping of the
Lanczos orthogonalisation may be considered as a clear sign of ‘non-
complexity’ of integrable systems.

The other data are

#### Krawtchouk

$\displaystyle B(x)=p(N-x),\quad D(x)=(1-p)x,\quad
0<p<1,\quad\eta(x)=x,\quad\mathcal{E}(n)=n,$ $\displaystyle
A_{n}=-p(N-n),\quad C_{n}=-(1-p)n,\quad\mu_{2}=2p(1-p)<1.$ (5.14)

#### Dual Hahn

($a,b>0$)

$\displaystyle B(x)=\frac{(x+a)(x+a+b-1)(N-x)}{(2x-1+a+b)(2x+a+b)},\quad
D(x)=\frac{x(x+b-1)(x+a+b+N-1)}{(2x-2+a+b)(2x-1+a+b)},$
$\displaystyle\eta(x)=x(x+a+b-1),\ \mathcal{E}(n)=n,\ A_{n}=-(n+a)(N-n),\
C_{n}=-n(b+N-n),$
$\displaystyle\mu_{2}=\frac{N+2}{10}\cdot\frac{4-5(a+b)+10ab-6N+5(a+b)N+2N^{2}}{2N+3(a+b)-2}.$
(5.15)

###  5.2 affine $q$-Krawtchouk and dual $q$-Hahn

These systems share the same functions $R_{0}$ and $R_{1}$

$\displaystyle
R_{0}(\mathcal{H})=(q^{-\tfrac{1}{2}}-q^{\tfrac{1}{2}})^{2}(\mathcal{H}^{\prime})^{2},\quad
R_{1}(\mathcal{H})=(q^{-\tfrac{1}{2}}-q^{\tfrac{1}{2}})^{2}\mathcal{H}^{\prime},\quad\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+1,\quad\mathcal{E}(n)=q^{-n}-1,$
$\displaystyle
R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-q)^{2}q^{-2n},$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-1)q^{-n},\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=(q-1)q^{-n}.$

For both systems $A_{n}$ and $C_{n}$ are quadratic polynomials in $q^{n}$ so
that $\mu_{2m}$ can be exactly calculated in a closed form.

#### affine $q$-Krawtchouk

($0<p<q^{-1}$)

$\displaystyle B(x)=(q^{x-N}-1)(1-pq^{x+1}),\quad
D(x)=pq^{x-N}(1-q^{x}),\quad\mathcal{E}(n)=q^{-n}-1,$ $\displaystyle
A_{n}=-(q^{n-N}-1)(1-pq^{n+1}),\quad C_{n}=-pq^{n-N}(1-q^{n}),\quad\
\eta(x)=q^{-x}-1,$
$\displaystyle\mu_{2m}=2p(q^{-1}-1)^{2m}\sum_{n=0}^{N-1}q^{n+1-N}(q^{n-N}-1)(1-pq^{n+1})(1-q^{n+1})q^{-2mn}/|\\!|\eta|\\!|^{2},$
(5.16)
$\displaystyle\qquad|\\!|\eta|\\!|^{2}=N+\frac{q^{-2N}\bigl{(}1-q^{N})(1-q^{N}(1-2q)\bigr{)}}{1-q^{2}}.$
(5.17)

#### dual $q$-Hahn

($0<a,b<1$)

$\displaystyle
B(x)=\frac{(q^{x-N}-1)(1-aq^{x})(1-abq^{x-1})}{(1-abq^{2x-1})(1-abq^{2x})},$
$\displaystyle
D(x)=aq^{x-N-1}\frac{(1-q^{x})(1-abq^{x+N-1})(1-bq^{x-1})}{(1-abq^{2x-2})(1-abq^{2x-1})},$
$\displaystyle\mathcal{E}(n)=q^{-n}-1,\quad\eta(x)=(q^{-x}-1)(1-abq^{x-1}),$
$\displaystyle A_{n}=-(1-aq^{n})(q^{n-N}-1),\quad
C_{n}=-aq^{-1}(1-q^{n})(q^{n-N}-b),$
$\displaystyle\mu_{2m}=2aq^{-1}(q^{-1}-1)^{2m}\sum_{n=0}^{N-1}(1-aq^{n})(q^{n-N}-1)(1-q^{n+1})(q^{n+1-N}-b)q^{-2mn}/|\\!|\eta|\\!|^{2}.$
(5.18)

###  5.3 quantum $q$-Krawtchouk

The system has the functions $R_{0}$ and $R_{1}$

$\displaystyle
R_{0}(\mathcal{H})=(q^{-\tfrac{1}{2}}-q^{\tfrac{1}{2}})^{2}(H^{\prime})^{2},\quad
R_{1}(\mathcal{H})=(q^{-\tfrac{1}{2}}-q^{\tfrac{1}{2}})^{2}H^{\prime},\quad\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}-1,\quad\mathcal{E}(n)=1-q^{n},$
$\displaystyle
R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-q)^{2}q^{2n},$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=(1-q)q^{n},\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(q^{-1}-1)q^{n}.$

Other data are ($p>q^{-N}$)

$\displaystyle B(x)=p^{-1}q^{x}(q^{x-N}-1),\quad
D(x)=(1-q^{x})(1-p^{-1}q^{x-N-1}),\ \mathcal{E}(n)=1-q^{n},$ $\displaystyle
A_{n}=-p^{-1}q^{-n-N-1}(1-q^{N-n}),\quad
C_{n}=-(q^{-n}-1)(1-p^{-1}q^{-n}),\quad\eta(x)=q^{-x}-1,$

The $A_{n}$ and $C_{n}$ are quadratic polynomials in $q^{-n}$ so that
$\mu_{2m}$ can be exactly calculated in a closed form,

$\displaystyle\mu_{2m}=2p^{-1}(1-q)^{2m}\sum_{n=0}^{N-1}q^{-n-N-1}(1-q^{N-n})(q^{-n-1}-1)(1-p^{-1}q^{-n-1})q^{2mn}/|\\!|\eta|\\!|^{2},$
(5.19)
$\displaystyle\qquad|\\!|\eta|\\!|^{2}=N+\frac{q^{-2N}\bigl{(}1-q^{N})(1-q^{N}(1-2q)\bigr{)}}{1-q^{2}}.$

###  5.4  $q$–Krawtchouk

The system has the functions $R_{0}$ and $R_{1}$ with
$\mathcal{E}(n)=(q^{-n}-1)(1+pq^{n})$ ($p>0$),

$\displaystyle
R_{0}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\bigl{(}\mathcal{H}^{\prime\,2}+p(q^{-\frac{1}{2}}+q^{\frac{1}{2}})^{2}\bigr{)},\quad
R_{1}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\mathcal{H}^{\prime},\quad\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+1-p,$
$\displaystyle
R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-q)^{2}(q^{-n}+pq^{n})^{2},$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-1)(q^{-n}+pq^{n+1})\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(1-q)(q^{-n}+pq^{n-1}).$

Other data are ($p>0$)

$\displaystyle B(x)=q^{x-N}-1,\quad
D(x)=p(1-q^{x}),\quad\mathcal{E}(n)=(q^{-n}-1)(1+pq^{n}),\quad\eta(x)=q^{-x}-1,$
$\displaystyle
A_{n}=-\frac{(q^{n-N}-1)(1+pq^{n})}{(1+pq^{2n})(1+pq^{2n+1})}\,,\quad
C_{n}=-pq^{2n-N-1}\frac{(1-q^{n})(1+pq^{n+N})}{(1+pq^{2n-1})(1+pq^{2n})}.$

As $A_{n}$ and $C_{n}$ are rational function of $q^{n}$, exact calculation of
$\mu_{2m}$ is rather complicated,

$\displaystyle\mu_{2m}=\frac{2p(q^{-1}-1)^{2m}}{|\\!|\eta|\\!|^{2}}\sum_{n=0}^{N-1}q^{2n-N+1}\frac{(q^{n-N}-1)(1+pq^{n})}{(1+pq^{2n})(1+pq^{2n+1})}$
$\displaystyle\hskip
128.0374pt\times\frac{(1-q^{n+1})(1+pq^{n+1+N})}{(1+pq^{2n+1})(1+pq^{2n+2})}(q^{-n}+pq^{n+1})^{2m}$
(5.20)
$\displaystyle\qquad|\\!|\eta|\\!|^{2}=N+\frac{q^{-2N}\bigl{(}1-q^{N})(1-q^{N}(1-2q)\bigr{)}}{1-q^{2}}.$

###  5.5 Hahn and Racah

These two systems have similar structures.

#### Hahn

The system has the functions $R_{0}$ and $R_{1}$ with
$\mathcal{E}(n)=n(n+a+b-1)$ ($a,b>0$),

$\displaystyle R_{0}(\mathcal{H})=4\mathcal{H}+(a+b-2)(a+b),\quad R_{1}=2,$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=2n+a+b,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(2n+a+b-2).$

Other data are

$\displaystyle B(x)=(x+a)(N-x),\quad D(x)=x(b+N-x),\quad\eta(x)=x,$
$\displaystyle A_{n}=-\frac{(n+a)(n+a+b-1)(N-n)}{(2n-1+a+b)(2n+a+b)},\quad
C_{n}=-\frac{n(n+b-1)(n+a+b+N-1)}{(2n-2+a+b)(2n-1+a+b)}.$

As $A_{n}$ and $C_{n}$ are rational functions of $n$, exact calculation of
$\mu_{2m}$ is rather complicated,

$\displaystyle\mu_{2m}=\frac{2}{|\\!|\eta|\\!|^{2}}\sum_{n=0}^{N-1}\frac{(n+a)(n+a+b-1)(N-n)}{(2n-1+a+b)}\cdot\frac{(n+1)(n+b)(n+a+b+N)}{(2n+1+a+b)}$
$\displaystyle\hskip 142.26378pt\times(2n+a+b)^{2(m-1)}$ (5.21)
$\displaystyle\qquad\qquad\qquad|\\!|\eta|\\!|^{2}=\frac{N(N+1)(2N+1)}{6}.$

#### Racah

The system has the functions $R_{0}$ and $R_{1}$ with
$\mathcal{E}(n)=n(n+\tilde{d})$ ($a,b>0$),

$\displaystyle R_{0}(\mathcal{H})=4\mathcal{H}+\tilde{d}^{2}-1\quad
R_{1}=2,\quad\tilde{d}\stackrel{{\scriptstyle\text{def}}}{{=}}a+b-N-d-1,\quad\eta(x)=x(x+d),$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=2n+\tilde{d}+1,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(2n+\tilde{d}-1).$

Other data are ($d>0,\ a>N+d,\ 0<b<1+d$),

$\displaystyle B(x)=-\frac{(x+a)(x+b)(x-N)(x+d)}{(2x+d)(2x+d+1)},\quad
D(x)=-\frac{(x+d-a)(x+d-b)(x+d+N)x}{(2x+d-1)(2x+d)},$ $\displaystyle
A_{n}=\frac{(n+a)(n+b)(n-N)(n+\tilde{d})}{(2n+\tilde{d})(2n+\tilde{d}+1)},\quad
C_{n}=\frac{(n+\tilde{d}-a)(n+\tilde{d}-b)(n+\tilde{d}+N)n}{(2n+\tilde{d}-1)(2n+\tilde{d})}.$

As $A_{n}$ and $C_{n}$ are rational functions of $n$, exact calculation of
$\mu_{2m}$ is rather complicated,

$\displaystyle\mu_{2m}=\frac{2}{|\\!|\eta|\\!|^{2}}\sum_{n=0}^{N-1}\frac{(n+a)(n+b)(n-N)(n+\tilde{d})}{(2n+\tilde{d})}\times(2n+\tilde{d}+1)^{2(m-1)}$
$\displaystyle\hskip
85.35826pt\times\frac{(n+1)(n+1+\tilde{d}-a)(n+1+\tilde{d}-b)(n+1+\tilde{d}+N)}{(2n+\tilde{d}+2)},$
(5.22)
$\displaystyle\qquad\qquad\qquad|\\!|\eta|\\!|^{2}=\sum_{x=0}^{N}x^{2}(x+d)^{2}.$

###  5.6 $q$-Hahn and $q$-Racah

These two systems have similar structures.

####  $q$-Hahn

The system has the functions $R_{0}$ and $R_{1}$ with
$\mathcal{E}(n)=(q^{-n}-1)(1-abq^{n-1})$,

$\displaystyle
R_{0}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\bigl{(}\mathcal{H}^{\prime\,2}-ab(1+q^{-1})^{2}\bigr{)},\
R_{1}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\mathcal{H}^{\prime},\
\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+1+abq^{-1},$
$\displaystyle
R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-q)^{2}(q^{-n}-abq^{n-1})^{2},$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-1)(q^{-n}-abq^{n}),\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(1-q)(q^{-n}-abq^{n-2}).$

Other data are ($0<a,b<1$),

$\displaystyle B(x)=(1-aq^{x})(q^{x-N}-1),\quad
D(x)=aq^{-1}(1-q^{x})(q^{x-N}-b),\quad\eta(x)=q^{-x}-1,$ $\displaystyle
A_{n}=-\frac{(q^{n-N}-1)(1-aq^{n})(1-abq^{n-1})}{(1-abq^{2n-1})(1-abq^{2n})},$
$\displaystyle
C_{n}=-aq^{n-N-1}\,\frac{(1-q^{n})(1-abq^{n+N-1})(1-bq^{n-1})}{(1-abq^{2n-2})(1-abq^{2n-1})},$

As $A_{n}$ and $C_{n}$ are rational function of $q^{n}$, exact calculation of
$\mu_{2m}$ is rather complicated,

$\displaystyle\mu_{2m}=\frac{2a(q^{-1}-1)^{2m}}{|\\!|\eta|\\!|^{2}}\sum_{n=0}^{N-1}q^{-2mn+n-N}\frac{(q^{n-N}-1)(1-aq^{n})(1-abq^{n-1})}{(1-abq^{2n-1})}\cdot(1-abq^{2n})^{2(m-1)}$
$\displaystyle\hskip
128.0374pt\times\frac{(1-q^{n+1})(1-abq^{n+N})(1-bq^{n})}{(1-abq^{2n+1})},$
(5.23)
$\displaystyle\qquad\qquad\qquad|\\!|\eta|\\!|^{2}=\sum_{x=0}^{N}(q^{-x}-1)^{2}.$

####  $q$-Racah

The system has the functions $R_{0}$ and $R_{1}$ with
$\mathcal{E}(n)=(q^{-n}-1)(1-\tilde{d}q^{n})$,

$\displaystyle
R_{0}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\bigl{(}\mathcal{H}^{\prime\,2}-(q^{-\frac{1}{2}}+q^{\frac{1}{2}})^{2}\tilde{d}\bigr{)},\
R_{1}(\mathcal{H})=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})^{2}\mathcal{H}^{\prime},\
\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+1+\tilde{d},$
$\displaystyle
R_{1}\bigl{(}\mathcal{E}(n)\bigr{)}^{2}+4R_{0}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-q)^{2}(q^{-n}-\tilde{d}q^{n})^{2},$
$\displaystyle\Longrightarrow\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=(q^{-1}-1)(q^{-n}-\tilde{d}q^{n+1}),\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(1-q)(q^{-n}-\tilde{d}q^{n-1}).$

Other data are
($\tilde{d}\stackrel{{\scriptstyle\text{def}}}{{=}}abd^{-1}q^{-N-1}$, $0<d<1,\
0<a<q^{N}d,\ qd<b<1$),

$\displaystyle
B(x)=-\frac{(1-aq^{x})(1-bq^{x})(1-q^{x-N})(1-dq^{x})}{(1-dq^{2x})(1-dq^{2x+1})},\quad\eta(x)=(q^{-x}-1)(1-dq^{x}),$
$\displaystyle
D(x)=-\tilde{d}\,\frac{(1-a^{-1}dq^{x})(1-b^{-1}dq^{x})(1-dq^{N+x})(1-q^{x})}{(1-dq^{2x-1})(1-dq^{2x})},$
$\displaystyle
A_{n}=\,\frac{(1-aq^{n})(1-bq^{n})(1-q^{n-N})(1-\tilde{d}q^{n})}{(1-\tilde{d}q^{2n})(1-\tilde{d}q^{2n+1})},$
$\displaystyle
C_{n}=d\,\frac{(1-a^{-1}\tilde{d}q^{n})(1-b^{-1}\tilde{d}q^{n})(1-\tilde{d}q^{n+N})(1-q^{n})}{(1-\tilde{d}q^{2n-1})(1-\tilde{d}q^{2n})}.$

As $A_{n}$ and $C_{n}$ are rational function of $q^{n}$, exact calculation of
$\mu_{2m}$ is rather complicated,

$\displaystyle\mu_{2m}=\frac{2d(q^{-1}-1)^{2m}}{|\\!|\eta|\\!|^{2}}\sum_{n=0}^{N-1}q^{-2mn}\frac{(1-aq^{n})(1-bq^{n})(1-q^{n-N})(1-\tilde{d}q^{n})}{(1-\tilde{d}q^{2n})}\cdot(1-\tilde{d}q^{2n+1})^{2(m-1)}$
$\displaystyle\hskip
128.0374pt\times\frac{(1-a^{-1}\tilde{d}q^{n+1})(1-b^{-1}\tilde{d}q^{n+1})(1-\tilde{d}q^{n+1+N})(1-q^{n+1})}{(1-\tilde{d}q^{2n+2})},$
(5.24)
$\displaystyle\qquad\qquad\qquad|\\!|\eta|\\!|^{2}=\sum_{x=0}^{N}(q^{-x}-1)^{2}(1-dq^{x})^{2}.$

##  6 Evaluation of $\mu_{m}$ of other exactly solvable quantum mechanical
systems

In order to evaluate the moments $\\{\mu_{m}\\}$ for verifications of Krylov
complexity of general quantum mechanical systems through orthonormalisation of
a Krylov subspace, an appropriate definition of the inner product of operators
is essential. The simplest trace one
$(\mathcal{V},\mathcal{W})=\text{Tr}[\\![\mathcal{V}^{\dagger}\mathcal{W}]\\!]$
(2.8) is obviously ill-defined for most operators, which are unbounded. Here I
tentatively adopt the one including the finite temperature ($T$) effects [5],

$(\mathcal{V},\mathcal{W})_{\beta}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{Z}\text{Tr}[\\![e^{-\beta\mathcal{H}/2}\mathcal{V}^{\dagger}e^{-\beta\mathcal{H}/2}\mathcal{W}]\\!]=\frac{1}{Z}\sum_{n=0}^{\infty}\langle
n\left|e^{-\beta\mathcal{H}/2}\mathcal{V}^{\dagger}e^{-\beta\mathcal{H}/2}\mathcal{W}\right|n\rangle$
(6.1)

in which

$Z\stackrel{{\scriptstyle\text{def}}}{{=}}\text{Tr}[\\![e^{-\beta\mathcal{H}}]\\!],\quad\beta=1/T.$

However, as shown later in the examples §6.1–§6.3, the factor $Z$ cancels out
in the calculation of the moments. The finite temperature inner product
suppresses the contributions of higher energy eigenstates and make the trace
summable. The norm of an operator is defined by

$(\mathcal{V},\mathcal{W})_{\beta}=(\mathcal{W},\mathcal{V})_{\beta}^{*}\rightarrow|\\!|\mathcal{V}|\\!|_{\beta}^{2}\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{V},\mathcal{V})_{\beta},$
(6.2)

With this inner product the flip property
$(\mathcal{V},\mathcal{L}\mathcal{W})_{\beta}=(\mathcal{L}\mathcal{V},\mathcal{W})_{\beta}$
(2.10) holds but $\mathcal{V}^{\dagger}=\pm\mathcal{V}$ does not mean
$(\mathcal{V},\mathcal{L}\mathcal{V})_{\beta}=0$. The Lanczos
orthonormalisation reviewed in section 2 must be modified as follows [3]. At
each step, the $\mathcal{O}_{k}$ component in $\mathcal{L}\mathcal{O}_{k}$
must be subtracted. One starts with

$\displaystyle\mathcal{O}_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}\eta/|\\!|\eta|\\!|_{\beta},\quad(\mathcal{O}_{0},\mathcal{O}_{0})_{\beta}=1,$
(6.3)
$\displaystyle\mathcal{W}_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{0}-a_{0}\mathcal{O}_{0},\quad
a_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{O}_{0},\mathcal{L}\mathcal{O}_{0})_{\beta}\quad
b_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{0}|\\!|_{\beta}\quad\mathcal{O}_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{1}}\mathcal{W}_{0}.$
(6.4)

Now $\mathcal{O}_{1}$ is orthogonal with the previous $\mathcal{O}_{0}$,

$(\mathcal{O}_{j},\mathcal{O}_{l})_{\beta}=\delta_{j\,l},\quad j,l=0,1.$ (6.5)

The next step is the orthogonalisation of $\mathcal{L}\mathcal{O}_{1}$ with
$\mathcal{O}_{0}$,

$\mathcal{W}_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{1}-a_{1}\mathcal{O}_{1}-b_{1}\mathcal{O}_{0},\quad
a_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{O}_{1},\mathcal{L}\mathcal{O}_{1})_{\beta}\quad
b_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{1}|\\!|_{\beta},\quad\mathcal{O}_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{2}}\mathcal{W}_{1}.$
(6.6)

The orthogonality $(\mathcal{O}_{1},\mathcal{W}_{1})_{\beta}=0$ is obvious. By
construction, the orthogonality $(\mathcal{O}_{0},\mathcal{W}_{1})_{\beta}=0$
holds. As

$\displaystyle(\mathcal{O}_{0},\mathcal{W}_{1})_{\beta}$
$\displaystyle=(\mathcal{O}_{0},\mathcal{L}\mathcal{O}_{1})_{\beta}-b_{1}(\mathcal{O}_{0},\mathcal{O}_{0})_{\beta}=(\mathcal{L}\mathcal{O}_{0},\mathcal{O}_{1})_{\beta}-b_{1}=\frac{1}{b_{1}}(\mathcal{W}_{0},\mathcal{W}_{0})_{\beta}-b_{1}=0,$

the orthonormality up to 2 is established

$(\mathcal{O}_{j},\mathcal{O}_{l})_{\beta}=\delta_{j\,l},\quad j,l=0,1,2.$
(6.7)

The flip property
$(\mathcal{V},\mathcal{L}\mathcal{W})_{\beta}=(\mathcal{L}\mathcal{V},\mathcal{W})_{\beta}$
(2.10) plays an important role in the above and further calculations. The
process goes on as

$\mathcal{W}_{k}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}\mathcal{O}_{k}-a_{k}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k-1},\
a_{k}\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{O}_{k},\mathcal{L}\mathcal{O}_{k})_{\beta},\
b_{k+1}\stackrel{{\scriptstyle\text{def}}}{{=}}|\\!|\mathcal{W}_{k}|\\!|_{\beta},\
\mathcal{O}_{k+1}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{b_{k+1}}\mathcal{W}_{k}.$
(6.8)

It is easy to prove that $\mathcal{O}_{k+1}$ is orthogonal to all the previous
ones $(\mathcal{O}_{j},\mathcal{O}_{k+1})_{\beta}=0$, $j=0,1,\ldots,k$, by
assuming the previous ones are orthonormal

$(\mathcal{O}_{j},\mathcal{O}_{l})_{\beta}=\delta_{j\,l},\quad
j,l=0,1,\ldots,k.$ (6.9)

It is obvious by construction
$(\mathcal{O}_{k},\mathcal{W}_{k})_{\beta}=0=(\mathcal{O}_{k},\mathcal{O}_{k+1})_{\beta}$.
Likewise,

$\displaystyle(\mathcal{O}_{k-1},\mathcal{W}_{k})_{\beta}$
$\displaystyle=\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k-1}\bigr{)}_{\beta}=\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{O}_{k}\bigr{)}_{\beta}-b_{k}=\frac{1}{b_{k}}\bigl{(}\mathcal{O}_{k-1},\mathcal{L}\mathcal{W}_{k-1}\bigr{)}_{\beta}-b_{k}$
$\displaystyle=\frac{1}{b_{k}}\bigl{(}\mathcal{L}\mathcal{O}_{k-1},\mathcal{W}_{k-1}\bigr{)}_{\beta}-b_{k}=\frac{1}{b_{k}}\bigl{(}\mathcal{W}_{k-1}+b_{k-1}\mathcal{O}_{k-2},\mathcal{W}_{k-1}\bigr{)}_{\beta}-b_{k}=b_{k}-b_{k}=0,$

as
$(\mathcal{O}_{k-2},\mathcal{W}_{k-1})_{\beta}=b_{k}(\mathcal{O}_{k-2},\mathcal{O}_{k})_{\beta}=0$
by assumption. For $j\leq k-2$, the same logic goes

$\displaystyle(\mathcal{O}_{j},\mathcal{W}_{k})_{\beta}$
$\displaystyle=\bigl{(}\mathcal{O}_{j},\mathcal{L}\mathcal{O}_{k}-b_{k}\mathcal{O}_{k-1}\bigr{)}_{\beta}=(\mathcal{O}_{j},\mathcal{L}\mathcal{O}_{k})_{\beta}=\frac{1}{b_{k}}(\mathcal{L}\mathcal{O}_{j},\mathcal{W}_{k-1})_{\beta}$
$\displaystyle=\frac{1}{b_{k}}(\mathcal{W}_{j}+b_{j}\mathcal{O}_{j-1},\mathcal{W}_{k-1})_{\beta}=0,$

and the induction is complete. $\square$

The Hamiltonian $\mathcal{H}$ and the sinusoidal coordinate $\eta$ of the
exactly solvable quantum mechanical systems demonstrated in [15, 16, 14, 20]
all satisfy the same relationship

$[\mathcal{H},[\mathcal{H},\eta]\,]=\eta\,R_{0}(\mathcal{H})+[\mathcal{H},\eta]\,R_{1}(\mathcal{H})+R_{-1}(\mathcal{H}),$
(4.1)

therefore the essential formulas

$\displaystyle e^{it\mathcal{H}}\eta\,e^{-it\mathcal{H}}$
$\displaystyle=\mathcal{L}\eta\cdot\frac{e^{i\alpha_{+}(\mathcal{H})t}-e^{i\alpha_{-}(\mathcal{H})t}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})}-R_{-1}(\mathcal{H})/R_{0}(\mathcal{H})$
$\displaystyle\quad+\bigl{(}\eta+R_{-1}(\mathcal{H})/R_{0}(\mathcal{H})\bigr{)}\frac{-\alpha_{-}(\mathcal{H})e^{i\alpha_{+}(\mathcal{H})t}+\alpha_{+}(\mathcal{H})e^{i\alpha_{-}(\mathcal{H})t}}{\alpha_{+}(\mathcal{H})-\alpha_{-}(\mathcal{H})}.$
(4.14)

and

$\displaystyle\mathcal{L}^{m}\eta=$
$\displaystyle\,\eta\,\mathcal{A}_{m}(\mathcal{H})+\mathcal{L}\eta\,\mathcal{B}_{m}(\mathcal{H})+\mathcal{C}_{m}(\mathcal{H}),\qquad
m\in\mathbb{N}_{\geq 0},$ (4.4)

for the evaluation of the moments $\\{\mu_{m}\\}$ hold with the same
coefficients $\mathcal{A}_{m}(\mathcal{H})$ (4.5),
$\mathcal{B}_{m}(\mathcal{H})$ (4.6) and $\mathcal{C}_{m}(\mathcal{H})$ (4.7).

Due to the property $(\mathcal{V},\mathcal{L}\mathcal{V})_{\beta}\neq 0$ of
the finite temperature trace, odd moments as well as even moments

$\displaystyle\mu_{m}=(\mathcal{O}_{0},\mathcal{L}^{m}\mathcal{O}_{0})_{\beta}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{Z|\\!|\eta|\\!|_{\beta}^{2}}\text{Tr}[\\![e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta]\\!]$ (6.10)

need be evaluated. Since the denominator $Z|\\!|\eta|\\!|_{\beta}^{2}$ is
independent of $m$, the numerator only

$\tilde{\mu}_{m}\stackrel{{\scriptstyle\text{def}}}{{=}}\text{Tr}[\\![e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta]\\!]=\sum_{n=0}^{\infty}\langle
n\left|e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta\right|n\rangle$ (6.11)

will be discussed henceforth.

The orthonormal eigenvectors of the Hamiltonian $\mathcal{H}$ $|n\rangle$ have
the same structure as (3.3) and (3.4). But the forms of the sinusoidal
coordinates $\eta(x)$ and the energy spectrum $\mathcal{E}(n)$ are not
restricted to those listed in (3.5) and (3.6). As the types of the
eigenpolynomials $\\{P_{n}(\eta)\\}$ are more varied than those of the
discrete quantum mechanics, the uniform normalisatin condition $P_{n}(0)=1$
(3.7) no longer applies and the three term recurrence relation has a more
general form

$\displaystyle\eta P_{n}(\eta)$
$\displaystyle=A_{n}P_{n+1}(\eta)+B_{n}P_{n}(\eta)+C_{n}P_{n-1}(\eta).$ (6.12)

This simply means that $A_{n}+C_{n}$ in the simplification of the moments in
section 5 should be replaced by $-B_{n}$.

The simplification of $\langle n\left|e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta\right|n\rangle$ goes in parallel
with that in section 5 as follows

$\displaystyle\langle n\left|e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta\right|n\rangle=\sum_{l=n,n\pm
1}e^{-\beta(\mathcal{E}(n)+\mathcal{E}(l))/2}\langle n|\eta|l\rangle\langle
l|\mathcal{L}^{m}\eta|n\rangle.$ (6.13)

The three parts (a) $l=n$, (b) $l=n+1$ and (c) $l=n-1$ ($n\geq 1$) are
separately simplified.

#### (a) $l=n$ part

This part vanishes as shown in section 5

$\displaystyle\langle n|\eta|n\rangle\langle
n|\mathcal{L}^{m}\eta|n\rangle=\langle n|\eta|n\rangle\left\\{\langle
n|\eta|n\rangle\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\langle
n|\mathcal{L}\eta|n\rangle\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\mathcal{C}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\right\\},$

as $\langle n|\eta|n\rangle^{2}\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$
cancels $\langle n|\eta|n\rangle\mathcal{C}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}$
part and $\langle n|\mathcal{L}\eta|n\rangle=0$.

#### (b) $l=n+1$ part

This part is

$\displaystyle\langle n|\eta|n+1\rangle\langle
n+1|\mathcal{L}^{m}\eta|n\rangle$ $\displaystyle=\langle
n|\eta|n+1\rangle\left\\{\langle
n+1|\eta|n\rangle\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\langle
n+1|\mathcal{L}\eta|n\rangle\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\right\\}$
$\displaystyle=A_{n}C_{n+1}\left\\{\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\right\\}$
$\displaystyle=A_{n}C_{n+1}\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m},$
(6.14)

as in (5.9).

#### (b) $l=n-1$ part

This part is

$\displaystyle\langle n|\eta|n-1\rangle\langle
n-1|\mathcal{L}^{m}\eta|n\rangle$ $\displaystyle=\langle
n|\eta|n-1\rangle\left\\{\langle
n-1|\eta|n\rangle\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\langle
n-1|\mathcal{L}\eta|n\rangle\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\right\\}$
$\displaystyle=A_{n-1}C_{n}\left\\{\mathcal{A}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}+\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}\mathcal{B}_{m}\bigl{(}\mathcal{E}(n)\bigr{)}\right\\}$
$\displaystyle=A_{n-1}C_{n}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m},\quad
n\geq 1,$ (6.15)

as in (5.10). These lead to

$\displaystyle\langle n\left|e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta\right|n\rangle$
$\displaystyle=e^{-\beta(\mathcal{E}(n)+\mathcal{E}(n+1))/2}A_{n}C_{n+1}\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}+e^{-\beta(\mathcal{E}(n)+\mathcal{E}(n-1))/2}A_{n-1}C_{n}\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}^{m},$
(6.16)

and

$\displaystyle\tilde{\mu}_{m}$ $\displaystyle=\sum_{n=0}^{\infty}\langle
n\left|e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\mathcal{L}^{m}\eta\right|n\rangle$
$\displaystyle=2\sum_{n=0}^{\infty}e^{-\beta(\mathcal{E}(n)+\mathcal{E}(n+1))/2}A_{n}C_{n+1}\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}^{m}\times\left\\{\begin{array}[]{cr}1&m:\
\text{even}\\\ 0&m:\ \text{odd}\end{array}\right.$ (6.19)

The squared norm of the sinusoidal coordinate $\eta$ is simplified as before,

$\displaystyle Z|\\!|\eta|\\!|_{\beta}^{2}$
$\displaystyle=\text{Tr}[\\![e^{-\beta\mathcal{H}/2}\eta
e^{-\beta\mathcal{H}/2}\eta]\\!]$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{i=n,n\pm
1}e^{-\beta\mathcal{E}(n)/2}e^{-\beta\mathcal{E}(l)/2}\langle
n|\eta|l\rangle\langle l|\eta|n\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\left\\{e^{-\beta\mathcal{E}(n)}\langle
n|\eta|n\rangle\langle
n|\eta|n\rangle+e^{-\beta\bigl{(}\mathcal{E}(n)+\mathcal{E}(n+1)\bigr{)}/2}\langle
n|\eta|n+1\rangle\langle n+1|\eta|n\rangle\right.$
$\displaystyle=\sum_{n=0}^{\infty}\left\\{e^{-\beta\mathcal{E}(n)}B_{n}^{2}+2e^{-\beta\bigl{(}\mathcal{E}(n)+\mathcal{E}(n+1)\bigr{)}/2}A_{n}C_{n+1}\right\\}.$
(6.20)

These results are summarised as the following

###### Theorem 6.1

The moments of the exactly solvable ordinary quantum systems have a very
simple exact expression

$\mu_{2m}=\frac{2}{Z|\\!|\eta|\\!|_{\beta}^{2}}\sum_{n=0}^{N-1}e^{-\beta(\mathcal{E}(n)+\mathcal{E}(n+1))/2}A_{n}C_{n+1}\left(\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}\right)^{2m},\quad\mu_{2m-1}=0,\quad
m\in\mathbb{N}.$ (6.21)

The vanishing of the odd moments is the consequence of the exact solvability
of the Heisenberg equation of motion (4.4), (4.14). It applies to two non-
compact discrete quantum systems with the eigenpolynomials, the Meixner (M)
and Charlier (C), as well as the exactly solvable systems related to the
Hermite (H), Laguerre (L), Gegenbauer (G), and Jacobi (J) polynomials.

###### Remark 6.2

The vanishing of the odd moments $\mu_{2m-1}=0$ simply means that all
$a_{k}=(\mathcal{O}_{k},\mathcal{L}\mathcal{O}_{k})_{\beta}$ in (6.8) are also
vanishing. Therefore, the relationship between the Lanczos coefficients
$\\{b_{n}\\}$ and the moments $\\{\mu_{2m}\\}$ mentioned in Remark 2.4 is also
valid with the trace including the finite temperature effects (6.1).

In the rest of this section, the necessary data for the evaluation of the
moments $\mu_{2m}$ of the above six exactly solvable quantum mechanical
systems are provided. For the ordinary one-dimensional quantum systems, they
are the potential $U(x)$ in the Hamiltonian $\mathcal{H}=p^{2}+U(x)$ in which
$p$ is the canonical momentum operator conjugate to $x$, $[p,x]=-i$, the
sinusoidal coordinate $\eta(x)$, the energy eigenvalues $\mathcal{E}(n)$,
$R_{0}(\mathcal{H})$ and $R_{1}(\mathcal{H})$ for the derivation of
$\alpha_{\pm}\bigl{(}\mathcal{E}(n)\bigr{)}$ and the coefficients of the three
term recurrence relation $A_{n}$ $B_{n}$ and $C_{n}$.

###  6.1 Meixner and Charlier

These two are non-compact exactly solvable discrete quantum systems. For the
summability of the trace, the finite temperature effects (6.1) are needed.
These two systems have very simple structure

$\displaystyle M,C:\quad$ $\displaystyle\mathcal{E}(n)=n,\quad\eta(x)=x,$
$\displaystyle R_{0}=1,\quad R_{1}=0\ \Rightarrow\
\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=1,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-1.$

This simply means that all the moments are equal $\mu_{2m}=\mu_{2}$,
$m\in\mathbb{N}$. The Lanczos orthonormalisation stops at $\mathcal{O}_{2}$.
The other data are

$\displaystyle M:\quad$ $\displaystyle B(x)=\frac{c}{1-c}(x+b),\quad
D(x)=\frac{x}{1-c},\quad 0<c<1,\quad 0<b,$ $\displaystyle
A_{n}=-\frac{c}{1-c}(n+b),\quad C_{n}=-\frac{n}{1-c},$
$\displaystyle\mu_{2}=\frac{2c}{(1-c)^{2}Z|\\!|\eta|\\!|_{\beta}^{2}}\sum_{n=0}^{\infty}e^{-\beta(2n+1)/2}(n+1)(n+b),$
(6.22) $\displaystyle C:\quad$ $\displaystyle B(x)=a,\quad D(x)=x,\quad
A_{n}=-a,\quad C_{n}=-n,\quad 0<a,$
$\displaystyle\mu_{2}=\frac{2a}{Z|\\!|\eta|\\!|_{\beta}^{2}}\sum_{n=0}^{\infty}e^{-\beta(2n+1)/2}(n+1).$
(6.23)

As expected, the Boltzmann factor $e^{-\beta\mathcal{H}}$ introduced for the
trace definition in 6.1 makes the infinite sum of these two $\mu_{2}$’s
convergent. The summation can be carried out explicitly.

###  6.2 Hermite and Laguerre

These two are the best known examples of exactly solvable one-dimensional
quantum mechanical systems, the harmonic oscillator for the Hermite (H) and
the harmonic oscillator with a centrifugal potential for the Laguerre (L).
Like K and dH cases in §5.1, all the moments form geometrical sequences,
$\mu_{2m}=\lambda^{2(m-1)}\mu_{2}$, $\lambda=2$ for H and $\lambda=4$ for L.
The Krylov orthogonalisation stops at $\mathcal{O}_{2}$. The data for each
system are:

#### Hermite

$\displaystyle
U(x)=x^{2}-1,\quad-\infty<x<\infty,\quad\mathcal{E}(n)=2n,\quad\eta(x)=x,$
$\displaystyle R_{0}=4,\quad
R_{1}=0,\quad\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=2,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-2,$
$\displaystyle A_{n}=\frac{1}{2},\quad B_{n}=0,\quad C_{n}=n,$
$\displaystyle\Longrightarrow\mu_{2m}=2^{2m}.$ (6.24)

This is a very well-known result.

#### Laguerre

($g>1$)

$\displaystyle U(x)=x^{2}+g(g-1)/x^{2}-(1+2g),\quad
0<x<\infty,\quad\mathcal{E}(n)=4n,\quad\eta(x)=x^{2},$ $\displaystyle
R_{0}=16,\quad
R_{1}=0,\quad\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=4,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-4,$
$\displaystyle A_{n}=-(n+1),\quad B_{n}=2n+g+1/2,\quad C_{n}=-(n+g-1/2),$
$\displaystyle\mu_{2m}=\frac{2^{4m+1}}{Z|\\!|\eta|\\!|_{\beta}^{2}}\sum_{n=0}^{\infty}e^{-2\beta(2n+1)}(n+1)(n+g+1/2).$
(6.25)

As mentioned above, the Heisenberg operator solution for $x^{2}+1/x^{2}$ (L)
and $1/\sin^{2}x$ (G) potentials are reported by Nieto and Simmons [19].

###  6.3 Gegenbauer and Jacobi

These two systems seem to provide very interesting materials for verifying
Krylov complexity. They are defined on a finite line segment, $0<x<\pi$ for G
and $0<x<\pi/2$ for J. They both have quadratic energy spectrum in $n$. The
expression of $\mu_{2m}$ (6.21) in the Theorem 6.1 applies. Here are their
data:

#### Gegenbauer

($g>1$)

$\displaystyle U(x)=\frac{g(g-1)}{\sin^{2}x}-g^{2},\quad
0<x<\pi,\quad\mathcal{E}(n)=n(n+2g),\quad\eta(x)=\cos x,$ $\displaystyle
R_{0}(\mathcal{H})=4\mathcal{H}^{\prime}-1,\quad\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+g^{2},\quad
R_{1}=2,\quad\alpha_{\pm}(\mathcal{H})=1\pm 2\sqrt{\mathcal{H}^{\prime}},$
$\displaystyle\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=2(n+g)+1,\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-(2n+2g-1),$
$\displaystyle A_{n}=\frac{n+1}{2(n+g)},\quad B_{n}=0,\quad
C_{n}=\frac{n+2g-1}{2(n+g)}.$

#### Jacobi

($g,h>1$)

$\displaystyle
U(x)=\frac{g(g-1)}{\sin^{2}x}+\frac{h(h-1)}{\cos^{2}x}-(g+h)^{2},\quad
0<x<\pi/2,$ $\displaystyle\mathcal{E}(n)=4n(n+g+h),\quad\eta(x)=\cos 2x,$
$\displaystyle
R_{0}(\mathcal{H})=16(\mathcal{H}^{\prime}-1),\quad\mathcal{H}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{H}+(g+h)^{2},\quad
R_{1}=8,\quad\alpha_{\pm}(\mathcal{H})=4\pm 4\sqrt{\mathcal{H}^{\prime}},$
$\displaystyle\alpha_{+}\bigl{(}\mathcal{E}(n)\bigr{)}=4(2n+g+h+1),\quad\alpha_{-}\bigl{(}\mathcal{E}(n)\bigr{)}=-4(2n+g+h-1),$
$\displaystyle A_{n}=\frac{2(n+1)(n+g+h)}{(2n+g+h)(2n+g+h+1)},\quad
B_{n}=\frac{(h-g)(g+h-1)}{(2n+g+h-1)(2n+g+h+1)},$ $\displaystyle
C_{n}=\frac{2(n+g-1/2)(n+h-1/2)}{(2n+g+h-1)(2n+g+h)}.$

##  7 Summary and comments

For a group of exactly solvable compact discrete quantum systems, the moments
of the operators in a Krylov subspace spanned by a Liouville operator
$\mathcal{L}:=[\mathcal{H},\cdot]$ and the sinusoidal coordinate $\eta$ are
evaluated explicitly. They provide the essential tool for measuring the growth
of operators evolving under Hamiltonian dynamics [1]. Understanding of the
complexity of exactly solvable quantum systems would reveal the nature of
quantum chaos by contrast. The moments of exactly solvable non-compact
discrete dynamics and ordinary one-dimensional quantum systems are also
evaluated explicitly by adopting the inner product involving the Boltzmann
factor. These exactly solvable discrete quantum mechanical systems can be
regarded as a very special type of matrix models.

There are many exactly solvable ordinary quantum mechanical systems having
finitely many discrete energy levels. Among them, for example, the Morse
potential and the soliton potential ($-1/\cosh^{2}x$) are exactly solvable in
the Heisenberg picture, too [16] and the formula
$\mathcal{L}^{m}\eta=\,\eta\,\mathcal{A}_{m}(\mathcal{H})+\mathcal{L}\eta\,\mathcal{B}_{m}(\mathcal{H})+\mathcal{C}_{m}(\mathcal{H})$
(4.4) is applicable. It is quite natural to expect that their ‘complexity’ is
qualitatively different from that of the systems having infinitely many energy
levels only. I cannot apply the Lanczos algorithm to such systems as I do not
know how to include the contribution of the continuous energy levels in the
operator inner product.

Another type of exactly solvable discrete quantum mechanical systems is also
known [16, 20] and they are also solvable in the Heisenberg picture. Their
eigenvectors contain the hypergeometric orthogonal polynomials of Askey
scheme, the Wilson, Askey-Wilson, continuous (dual) ($q$) Hahn, Meixner-
Pollaczek, Al-Salam-Chihara, continuous (big) $q$-Hermite, continuous
$q$-Jacobi (Laguerre) polynomials. It is expected that the moments of these
quantum systems can be evaluated explicitly in a similar manner.

Exact Heisenberg operator solutions are also known for a family of multi-
particle dynamics, the Calogero models based on any root systems [21]. It is a
good challenge to generalise the present method for multi-particle systems.

Four explicit examples of multivariate discrete orthogonal polynomials, the
multivariate Krawtchouk, Meixner and two types of Rahman polynomials, are
constructed recently by myself [22]. They are eigenvectors of respective
Hamiltonians having nearest neighbour interactions. Investigation of their
complexity through Krylov orthonormalisation would expose the contrast between
integrable and chaotic multi-particle dynamics.

One of the motivations of the present research is to display the main
ingredients of Krylov complexity with the explicit description of the
Hamiltonian $\mathcal{H}$ and the operator $\eta$, which is lacking in some
reports.

## Acknowledgements

R. S. thanks Jen-Chi Lee for the hospitality and vigorous discussions during
his visit to National Yang Ming Chiao Tung University. He also thanks Yi Yang
for opening his eyes to the topic of Krylov complexity.

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