# Trace formulas for the modified Mathieu equation

Leon A. Takhtajan

###### Abstract

For the radial and one-dimensional Schrödinger operator $H$ with growing
potential $q(x)$ we outline a method of obtaining the trace identities — an
asymptotic expansion of the Fredholm determinant $\mathrm{det}_{F}(H-\lambda
I)$ as $\lambda\to-\infty$. As an illustrating example, we consider
Schrödinger operator with the potential $q(x)=2\cosh 2x$, associated with the
modified Mathieu equation.

Mathematics Subject Classification 2020. Primary 47E05; Secondary 34E05

Keywords. Fredholm determinant, Green-Liouville method, Mathieu equation,
radial and one-dimensional Schrödinger operator, Riccati equation, trace
identities

††Leon A. Takhtajan<EMAIL_ADDRESS>
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651,
USA
Euler International Mathematical Institute, Pesochnaya Nab. 10, Saint
Petersburg 197022, Russia

To my friend Ari Laptev on the occasion of his 70th birthday

##  1 Introduction

Mathieu functions, solutions of the Mathieu equation and modified Mathieu
equation, have many applications in mathematics and theoretical physics. In
particular, the modified Mathieu differential equation

$\frac{d^{2}\psi}{dx^{2}}+(\lambda-2\cosh 2x)\psi=0$

on the real axis $-\infty<x<\infty$, is a Schrödinger equation

$H\psi=\lambda\psi$

with the Hamiltionian

$H=-\frac{d^{2}}{dx^{2}}+2\cosh 2x$ (1.1)

in the Hilbert space $L^{2}({\mathbb{R}})$.

A positive, self-adjoint operator $H$ given by (1.1) is a Hamiltonian of the
quantum two-particle periodic Toda chain (after separation of the center of
mass), and has a purely discrete spectrum. As its classical analog, the
quantum $N$-particle periodic Toda chain is also completely integrable, as was
shown by M. Gutzwiller [1] for $N=2,3$ and by E.K. Sklyanin [2] for general
$N$. For $N=2$ the exact quantization condition in [1] was obtained by
applying Floquet theory to the potential $q(x)=2\cosh 2x$ with pure imaginary
period, and is formulated in terms of infinite Hill determinants (see [3, Ch.
XIX]). Sklyanin’s method of quantum separation of variables was developed
further by Q. Pasquier and M. Gaudin [4], and by S. Kharchev and D. Lebedev
[5].

Recently, N.A. Nekrasov and S.L. Shatashvili [6] found a remarkable connection
between quantum integrable systems and $\mathcal{N}=2$ four-dimensional
supersymmetric Yang-Mills theories in the special $\Omega$-background. Among
other interesting results they showed, at the physical level of rigor, that
exact quantization conditions for the quantum $N$-particle periodic Toda chain
can be written in terms of the so-called effective twisted superpotential for
the low energy effective two-dimensional gauge theory. In particular, Nekrasov
and Shatashvili gave a new interpretation of Gutzwiller and Kharchev and
Lebedev quantization conditions.

Replacing the kinetic term $\bm{P}^{2}$ in the Hamiltonian $H$ by
$2\cosh\bm{P}$, where $\bm{P}=-i\dfrac{d}{dx}$ is the quantum-mechanical
momentum operator, one gets the following functional-difference operator on
$L^{2}({\mathbb{R}})$,

$\hat{H}=2\cosh\bm{P}+2\cosh 2\bm{Q},$ (1.2)

where $\bm{Q}$ is the quantum-mechanical position operator. The operator
$\hat{H}$ is a positive self-adjoint operator on $L^{2}({\mathbb{R}})$ with a
purely discrete spectrum, and $\hat{H}^{-1}$ is a trace class operator.
Operators of such type naturally appear in the theory of topological strings
as a quantization of mirror curves of non-compact Calabi-Yau threefolds,
called “topological strings/spectral theory (TS/ST) duality”. These operators
have a purely discrete spectrum, and their Fredholm determinants

$\mathrm{det}_{F}(\hat{H}-\lambda I)=\frac{\det(\hat{H}-\lambda
I)}{\det\hat{H}}=\prod_{n=1}^{\infty}\left(1-\frac{\lambda}{\lambda_{n}}\right),$
(1.3)

where $\lambda_{n}$ are the eigenvalues, are conjectured to encode deep
relations in the enumerative geometry of the corresponding Calabi-Yau
manifolds. We refer the reader to M. Mariño comprehensive survey [7] for the
precise description of the method, formulation of the main conjectures and
references. Developing this method, A. Grassi, J. Gu and M. Mariño in [8]
conjectured an exact formula for the Fredholm determinant of the modified
Mathieu operator in terms of the so-called “exact quantum periods”. Though in
certain special cases TS/ST conjectures can be confirmed by numerical
computations, their mathematical derivation seems to be out of reach.

Nevertheless, it is quite remarkable that operators of the type (1.2) can be
rigorously studied by the method developed by Ari Laptev more then twenty
years ago in [9]. Namely, it was proved in [10] (see also [11]) that these
functional-difference operators have a purely discrete spectrum and their
eigenvalue counting function satisfies the Weyl’s law.

The logarithmic derivative of the Fredholm determinant
$a(\lambda)=\mathrm{det}_{F}(\hat{H}-\lambda I)$ is

$-\frac{d}{d\lambda}\log a(\lambda)=\mathrm{Tr}(\hat{H}-\lambda
I)^{-1}=\sum_{n=1}^{\infty}\frac{1}{\lambda-\lambda_{n}}.$

As $\lambda\to-\infty$, it admits the asymptotic expansion

$\frac{d}{d\lambda}\log
a(\lambda)=\alpha_{0}(\nu)+\sum_{n=1}^{\infty}\frac{c_{n}}{\nu^{n}}+O(|\nu|^{-\infty}),$

where $\nu=\sqrt{-\lambda}$ and the function $\alpha_{0}(\nu)$ is determined
by the asymptotic of the eigenvalues (cf. [12, Theorem 3.1]). The coefficients
$c_{2k}$ represent regularized divergent series
$\sum_{n=1}^{\infty}\lambda_{n}^{k}$ and it is expected that they are
expressed in terms of the potential of the functional-difference operator
$\hat{H}$. Relations of such type are called _trace identities_.

The trace identities for the Sturm-Liouville operators were derived in the
classic papers by I.M. Gelfand and B.M. Levitan [13] and L.A. Dikii [12], for
the radial Schrödinger operator with rapidly decaying potential $q(x)$ — by
V.S. Buslaev and L.D. Faddeev in [14], and for the one-dimensional Schrödinger
operator with rapidly decaying potential $q(x)$ — by L.D. Faddeev and V.E.
Zakharov in [15] (see also [16]).

Though the trace identities for growing as $|x|\to\infty$ potential $q(x)$
have not been explicitly considered in the literature, a closed topic —
semiclassical expansion in a complex domain, illustrated by the case of the
homogeneous quartic harmonic oscillator — was investigated by A. Voros in [17,
18]. One should also mention the relative trace identities — the asymptotic
expansion of

$\mathrm{Tr}\left\\{\left(-\frac{d^{2}}{dx^{2}}+q(x)+r(x)-\lambda
I\right)^{-1}-\left(-\frac{d^{2}}{dx^{2}}+q(x)-\lambda I\right)^{-1}\right\\}$

as $\lambda\to-\infty$, where potential $q(x)$ grows as $|x|\to\infty$, and
function $r(x)$ is rapidly decaying. The interesting example $q(x)=x^{2}$ was
considered by A. Pushnitski and I. Sorrell in [19].

In the present paper we outline a simple method for the derivation of the
trace identities for the Schrödinger operator with the potential $q(x)$ that
rapidly grows to infinity as $|x|\to\infty$, and illustrate it by explicit
formulas for the case $q(x)=2\cosh 2x$, the modified Mathieu operator (1.1).
In Sect. 2.1 we briefly review the Liouville-Green method, which is used to
obtain the trace identities for the radial Schrödinger operator in Sect.
2.2.1, and for the one-dimensional Schrödinger operator in Sect. 2.2.2. In
Section 3 we illustrate our method on two examples: a well-known radial
Schrödinger operator with potential $q(x)=e^{2x}$ in Sect. 3.1, and modified
Mathieu operator (1.1) in Sect. 3.2. Our main result — the trace identities
for the modified Mathieu operator — is presented in Theorem 1. Finally, in
Section 4 we prove that modified Mathieu differential equation has a solution
that behaves like the modified Bessel function of the second kind as
$x\to\infty$.

##  2 General case

Here we consider the differential equation

$-\psi^{\prime\prime}+q(x)\psi=\lambda\psi$ (2.1)

one the half-line $0<x<\infty$ and on the real line $\infty<x<\infty$, where
$q(x)$ is a positive smooth function that grows to infinity as $x\to\infty$
(or $|x|\to\infty$).

###  2.1 The Liouville-Green method

When $\lambda<0$, decaying as $x\to\infty$ solution of (2.1) has the following
double asymptotic

$\psi_{1}(x,\lambda)=\frac{C_{1}(\lambda)}{\sqrt[4]{q(x)-\lambda}}\,e^{-\int_{0}^{x}\sqrt{q(s)-\lambda}\,ds}(1+\varepsilon_{1}(x,\lambda)),$
(2.2)

where in may cases

$|\varepsilon_{1}(x,\lambda)|\leq\frac{\tau(x)}{|\lambda|},\quad\tau(x)\to
0\quad\text{as}\quad x\to\infty.$

Similarly, solution of (2.1) that grows as $x\to\infty$ has the asymptotic

$\psi_{2}(x,\lambda)=\frac{C_{2}(\lambda)}{\sqrt[4]{q(x)-\lambda}}\,e^{\int_{0}^{x}\sqrt{q(s)-\lambda}\,ds}(1+\varepsilon_{2}(x,\lambda)).$
(2.3)

This follows from the classical Liouville-Green method, developed by F. Olver
[20] (see also [21]).

Put $Q(x,\lambda)=q(x)-\lambda$ and introduce

$\chi(x,\lambda)=\log\psi_{1}(x,\lambda)+\frac{1}{4}\log
Q(x,\lambda)+\int_{0}^{x}\sqrt{Q(s,\lambda)}\,ds-\log C_{1}(\lambda)$ (2.4)

and

$\sigma(x,\lambda)=\chi^{\prime}(x,\lambda)=\frac{\psi_{1}^{\prime}(x,\lambda)}{\psi_{1}(x,\lambda)}+\frac{Q^{\prime}(x,\lambda)}{4Q(x,\lambda)}+\sqrt{Q(x,\lambda)},$
(2.5)

where prime stands for the $x$-derivative. Denoting for brevity
$\sigma=\sigma(x,\lambda)$ and $Q=Q(x,\lambda)$, we see that the function
$\sigma$ satisfies the Riccati equation

$\displaystyle\sigma^{\prime}$
$\displaystyle=\frac{\psi_{1}^{\prime\prime}}{\psi_{1}}-\left(\frac{\psi_{1}^{\prime}}{\psi_{1}}\right)^{2}+\frac{Q^{\prime\prime}}{4Q}-\frac{1}{4}\left(\frac{Q^{\prime}}{Q}\right)^{2}+\frac{Q^{\prime}}{2\sqrt{Q}}$
$\displaystyle=Q-\left(\sigma-\frac{Q^{\prime}}{4Q}-\sqrt{Q}\right)^{2}+\frac{Q^{\prime\prime}}{4Q}-\frac{1}{4}\left(\frac{Q^{\prime}}{Q}\right)^{2}+\frac{Q^{\prime}}{2\sqrt{Q}},$

or

$\sigma^{\prime}=-\sigma^{2}+2\sqrt{Q}\sigma+\frac{Q^{\prime}}{2Q}\sigma+\frac{Q^{\prime\prime}}{4Q}-\frac{5}{16}\left(\frac{Q^{\prime}}{Q}\right)^{2}.$
(2.6)

###  2.2 Trace identities

The Riccati equation (2.6) can be used to obtain an asymptotic expansion of
$\sigma(x,\lambda)$ as $\lambda\to-\infty$ and get the trace identities for
the Schrödinger operator with growing as $x\to\infty$ potential $q(x)$ on
$L^{2}(0,\infty)$ and on $L^{2}({\mathbb{R}})$. In the latter case potential
$q(x)$ is assumed to be an even function.

####  2.2.1 Radial Schrödinger operator

Here we consider the Schrödinger operator $H$ on $L^{2}(0,\infty)$ with a
smooth positive potential $q(x)$ satisfying

$\int_{0}^{\infty}\frac{dx}{\sqrt{q(x)}}<\infty,$ (2.7)

supplemented with the boundary condition $\psi(0)=0$. Restricting further, we
consider potentials satisfying the inequality $q(x)\geq Cx^{2+\varepsilon}$
for some $\varepsilon>0$. It follows from the Weyl’s law that $H^{-1}$ is a
trace class operator, and its Fredholm determinant is an entire function of
order $1/2$.

For every $\lambda$ equation (2.1) for $0<x<\infty$ has a unique solution
$\psi(x,\lambda)$ with the asymptotic

$\psi(x,\lambda)=\frac{1}{\sqrt[4]{q(x)}}\,e^{-\int_{0}^{x}\sqrt{q(s)}\,ds}(1+o(1))\quad\text{as}\quad
x\to\infty.$ (2.8)

For fixed $x$ solution $\psi(x,\lambda)$ is an entire function of $\lambda$ of
order $1/2$ and $\psi(0,\lambda)$ is proportional to the Fredholm determinant
$a(\lambda)$ of the operator $H$,

$a(\lambda)=\frac{\psi(0,\lambda)}{\psi(0,0)}.$

Comparing equations (2.2) and (2.8) for $\lambda<0$, we see that for such
$\lambda$ solution $\psi(x,\lambda)$ has asymptotic (2.2) with

$C_{1}(\lambda)=e^{\int_{0}^{\infty}\big{(}\sqrt{Q(x,\lambda)}-\sqrt{q(x)}\big{)}dx}.$

It follows from (2.7) that the integral in this formula is convergent, so
$C_{1}(\lambda)$ is well-defined.

Putting in formula (2.4) $\lambda=-\nu^{2}$, we have

$\displaystyle\chi(x,\lambda)=\log\psi(x,\lambda)+\frac{1}{4}\log(q(x)+\nu^{2})+\int_{0}^{x}\sqrt{q(s)+\nu^{2}}\,ds$
$\displaystyle-\int_{0}^{\infty}\Big{(}\sqrt{q(s)+\nu^{2}}-\sqrt{q(s)}\Big{)}ds.$

From here we obtain

$\displaystyle\chi(0,\lambda)$ $\displaystyle=\log a(\lambda)-a_{0}(\lambda),$
$\displaystyle\lim_{x\to\infty}\chi(x,\lambda)$ $\displaystyle=0,$

where

$a_{0}(\lambda)=-\frac{1}{4}\log(q(0)+\nu^{2})+\int_{0}^{\infty}\Big{(}\sqrt{q(x)+\nu^{2}}-\sqrt{q(x)}\Big{)}dx-\log\psi(0,0)$

(cf. [18, Appendix A]). It is not difficult to show that $a_{0}(\lambda)$
admits an asymptotic expansion as $\lambda=-\nu^{2}\to\infty$,

$a_{0}(\lambda)=\alpha(\nu)+\sum_{n=1}^{\infty}\frac{\alpha_{n}}{\nu^{n}}+O(|\nu|^{-\infty}),$
(2.9)

where the leading term — a function $\alpha(\nu)$ — is determined by the
asymptotic of the eigenvalues and can be computed explicitly (see examples in
Section 3).

The function $\sigma(x,\lambda)=\chi^{\prime}(x,\lambda)$ satisfies the
Riccati equation (2.6), which admits the asymptotic solution

$\sigma(x,\lambda)=\sum_{n=1}^{\infty}\frac{c_{n}(x)}{\nu^{n}}+O(|\nu|^{-\infty}),$
(2.10)

where the coefficients $c_{n}(x)$ are determined recursively in terms of the
potential $q(x)$ and its derivatives. Since

$\log\chi(0,\lambda)=-\int_{0}^{\infty}\sigma(x,\lambda)dx,$ (2.11)

we have

$\log a(\lambda)=\alpha_{0}(\lambda)-\int_{0}^{\infty}\sigma(x,\lambda)dx.$
(2.12)

Using the asymptotic expansion (2.10), we obtain the trace identities

$\log
a(\lambda)=\alpha(\nu)+\sum_{n=1}^{\infty}\frac{c_{n}}{\nu^{n}}+O(|\nu|^{-\infty}),\quad\text{where}\quad
c_{n}=\alpha_{n}-\int_{0}^{\infty}c_{n}(x)dx.$ (2.13)

####  2.2.2 One-dimensional Schrödinger operator

Here we assume that the potential $q(x)$ is a smooth even function satisfying
(2.7). A fundamental system of solutions of the differential equation (2.1) is
given by solutions $\psi_{1}(x,\lambda)$ and $\psi_{2}(x,\lambda)$ with the
following asymptotic as $x\to\infty$,

$\displaystyle\psi_{1}(x,\lambda)$
$\displaystyle=\frac{1}{\sqrt[4]{q(x)}}\,e^{-\int_{0}^{x}\sqrt{q(s)}\,ds}(1+o(1)),$
(2.14) $\displaystyle\psi_{2}(x,\lambda)$
$\displaystyle=\frac{1}{\sqrt[4]{q(x)}}\,e^{\int_{0}^{x}\sqrt{q(s)}\,ds}(1+o(1)).$
(2.15)

Assuming that

$\lim_{x\to\infty}\frac{q^{\prime}(x)}{\big{(}\sqrt{q(x)}\big{)}^{3}}=0,$

we get from (2.14) and (2.15) that

$W(\psi_{1},\psi_{2})=2,$

where $W(f,g)=fg^{\prime}-f^{\prime}g$ is the Wronskian of two functions. The
functions $\psi_{1}(x,\lambda)$ and $\psi_{2}(x,\lambda)$ satisfy asymptotic
formulas (2.2)–(2.3), where

$C_{1}(\lambda)=e^{\int_{0}^{\infty}\big{(}\sqrt{Q(x,\lambda)}-\sqrt{q(x)}\big{)}dx}\quad\text{and}\quad
C_{2}(\lambda)=e^{-\int_{0}^{\infty}\big{(}\sqrt{Q(x,\lambda)}-\sqrt{q(x)}\big{)}dx}.$

Another fundamental system of solutions is given by the functions
$\psi_{1}(-x,\lambda)$ and $\psi_{2}(-x,\lambda)$, and we have

$\psi_{1}(x,\lambda)=t_{11}(\lambda)\psi_{1}(-x,\lambda)+t_{12}(\lambda)\psi_{2}(-x,\lambda),$
(2.16)

where

$t_{12}(\lambda)=\frac{1}{2}W(\psi_{1}(x,\lambda),\psi_{1}(-x,\lambda)).$
(2.17)

For fixed $x$ the functions $\psi_{1}(x,\lambda)$ and $\psi_{2}(x,\lambda)$
are entire functions of order $1/2$, as it can be shown using condition (2.7).
Therefore, $t_{12}(\lambda)$ is an entire function of order $1/2$ with zeros
at the eigenvalues of the operator $H$. As in the previous section, for the
Fredholm determinant $a(\lambda)$ of the operator $H$ we obtain

$a(\lambda)=\frac{t_{12}(\lambda)}{t_{12}(0)}.$

It follows from (2.2) that the function $\chi(x,\lambda)$, defined by (2.4),
satisfies

$\lim_{x\to\infty}\chi(x,\lambda)=0.$

To investigate its behavior as $x\to-\infty$ we observe that it follows from
(2.14)–(2.15) that as $x\to-\infty$ the first term in (2.16) is exponentially
small with respect to the second term, so

$\lim_{x\to-\infty}(\log\psi_{1}(x,\lambda)-\log\psi_{2}(-x,\lambda))=\log
t_{12}(\lambda).$

Since the antiderivative $\int_{0}^{x}\sqrt{Q(s,\lambda)}ds$ is an odd
function of $x$, we obtain

$\lim_{x\to-\infty}\chi(x,\lambda)=\log a(\lambda)-a_{0}(\lambda),$

where now

$a_{0}(\lambda)=\int_{-\infty}^{\infty}\Big{(}\sqrt{q(x)+\nu^{2}}-\sqrt{q(x)}\Big{)}dx-\log
t_{12}(0).$

As in Sect. 2.2.1, the function $a_{0}(\lambda)$ admits an asymptotic
expansion of the type (2.9). The corresponding Riccati equation (2.6) has an
asymptotic solution

$\sigma(x,\lambda)=\sum_{n=1}^{\infty}\frac{c_{n}(x)}{\nu^{n}}+O(|\nu|^{-\infty}),$

where the coefficients $c_{n}(x)$ are determined recursively and are expressed
in terms of the potential $q(x)$ and its derivatives. Thus we obtain the trace
identities

$\log
a(\lambda)=\alpha(\nu)+\sum_{n=1}^{\infty}\frac{c_{n}}{\nu^{n}}+O(|\nu|^{-\infty}),\quad\text{where}\quad
c_{n}=\alpha_{n}-\int_{-\infty}^{\infty}c_{n}(x)dx.$ (2.18)

######  Remark.

The interesting example of the harmonic oscillator — the potential
$q(x)=x^{2}$ — does not satisfy condition (2.7). However, one can modify the
outlined here method by exploiting the substitution
$x=\sqrt{-\lambda}t$.222This is the substitution used in [20] to obtain
uniform asymptotic for Weber functions (parabolic cylinder functions). In this
case the Fredholm determinant is

$a(\lambda)=\frac{2^{-\frac{\lambda}{2}}\sqrt{\pi}}{\Gamma(\frac{1-\lambda}{2})}$

and corresponding trace identities give an alternate derivation of the
Stirling asymptotic expansion for the Euler gamma-function (for an equivalent
approach, see [17, Part IV]).

##  3 Examples

###  3.1 Potential $q(x)=e^{2x}$ on the half-line

It is well-known that the differential equation

$-\psi^{\prime\prime}+e^{2x}\psi=\lambda\psi$

has two linearly independent solutions, the modified Bessel function of the
first kind $I_{\nu}(e^{x})$, and the modified Bessel function of the second
kind $K_{\nu}(e^{x})$, where $\nu=\sqrt{-\lambda}$. For fixed $\nu$ and real
$y\to\infty$ we have the asymptotic

$K_{\nu}(y)=\sqrt{\frac{\pi}{2y}}e^{-y}\left(1+O(y^{-1})\right),$ (3.1)

so the eigenvalues determined by the boundary condition $\psi(0)=0$ are the
zeros $\\{\lambda_{n}\\}_{n=1}^{\infty}$ of an entire function $K_{\nu}(1)$,
where $\lambda=-\nu^{2}$. The eigenvalues $\lambda_{n}$ are positive, simple
and accumulate to infinity, and

$a(\lambda)=\frac{K_{\nu}(1)}{K_{0}(1)}.$ (3.2)

######  Remark.

It is well-known (see, e.g., [22]) that the total number of zeros of
$K_{ik}(1)$ with $0<k<T$ is

$\frac{T}{\pi}\log\frac{2T}{e}+O(1),$

so for the eigenvalue counting function $N(\lambda)$ we obtain

$N(\lambda)=\frac{\sqrt{\lambda}}{\pi}\log\frac{2\sqrt{\lambda}}{e}+O(1).$

This also easily follows from the Weyl’s law. Thus as $n\to\infty$ for the
$n$-th eigenvalue $\lambda_{n}$ we have

$\lambda_{n}\simeq\left(\frac{2\pi n}{eW(\frac{2\pi n}{e})}\right)^{2},$

where $W(x)$ is the Lambert function, $W(x)e^{W(x)}=x$ for $x>0$.

The asymptotic expansion of $\log a(\lambda)$ as $\lambda=-\nu^{2}\to-\infty$
can be easily obtained from the well-known asymptotic of the modified Bessel
function of the second kind333One needs to put $z=1/\nu$ and
$p=\nu/\sqrt{1+\nu^{2}}$ in formula (7.17) in [20, Ch.10, §7].. However, it
can also be derived directly, using the method outlined in Sect. 2.2.1.

Namely, we have

$\displaystyle\int_{0}^{x}\sqrt{e^{2s}+\nu^{2}}\,ds=\sqrt{\nu^{2}+e^{2x}}+\nu
x-\nu\log\left(\nu+\sqrt{\nu^{2}+e^{2x}}\right)$
$\displaystyle-\sqrt{\nu^{2}+1}+\nu\log\left(\nu+\sqrt{\nu^{2}+1}\right).$

Since

$\displaystyle\sqrt{\nu^{2}+e^{2x}}+\nu
x-\nu\log\left(\nu+\sqrt{\nu^{2}+e^{2x}}\right)=e^{x}+O(e^{-x})\quad\text{as}\quad
x\to\infty,$

comparison with (2.2) gives $K_{\nu}(e^{x})=\psi_{1}(x,\lambda)$, where

$C_{1}(\lambda)=\sqrt{\frac{\pi}{2}}\,e^{\sqrt{\nu^{2}+1}-\nu\log\left(\nu-\sqrt{\nu^{2}+1}\right)}.$

Of course, this is a well-known asymptotic of the modified Bessel function of
the second kind,

$K_{\nu}(e^{x})=\sqrt{\frac{\pi}{2}}\frac{e^{-\\{\sqrt{\nu^{2}+e^{2x}}+\nu
x-\nu\log(\nu+\sqrt{\nu^{2}+e^{2x}})\\}}}{\sqrt[4]{\nu^{2}+e^{2x}}}(1+O(|\nu|^{-1})),$
(3.3)

which is uniform for $-\infty<x<\infty$ (see [20, Ch. 10, §7]).

Now the function $\chi(x,\lambda)$, defined in (2.4), satisifes

$\displaystyle\lim_{x\to\infty}\chi(x,\lambda)$ $\displaystyle=0,$
$\displaystyle\chi(0,\lambda)$ $\displaystyle=\log a(\lambda)-a_{0}(\lambda),$

where

$a_{0}(\lambda)=\sqrt{\nu^{2}+1}-\nu\log\left(\nu+\sqrt{\nu^{2}+1}\right)-\frac{1}{4}\log(\nu^{2}+1)+\frac{1}{2}\log\frac{\pi}{2}-\log
K_{0}(1).$

The function $a_{0}(\lambda)$ admits an asymptotic expansion (2.9), where

$\alpha_{0}(\lambda)=\nu\log(2\nu)-\nu-\frac{1}{2}\log\nu+\frac{1}{2}\log\frac{\pi}{2}-\log
K_{0}(1).$

This formula agrees with the asymptotic

$K_{\nu}(e^{x})=\sqrt{\frac{\pi}{2\nu}}\left(\frac{e^{x+1}}{2\nu}\right)^{\\!-\nu}(1+O(|\nu|^{-1}))$
(3.4)

for fixed $x$ and $\nu\to\infty$, and with the Weyl’s law.

The function $\sigma(x,\lambda)$, defined in (2.5), satisfies the Riccati
equation (2.6). Its asymptotic expansion (2.10) can be easily obtained by
using the substitution $\tau(t,\lambda)=\sigma(t+\log\nu,\lambda)$, which
transforms (2.6) into the equation

$\tau^{\prime}=-\tau^{2}+2\nu\sqrt{1+e^{2t}}\tau+\frac{e^{2t}}{1+e^{2t}}\tau+\frac{4e^{2t}-e^{4t}}{4(1+e^{2t})^{2}}.$
(3.5)

We have

$\tau(t,\nu)=\sum_{n=1}^{\infty}\frac{c_{n}(t)}{\nu^{n}},$

where

$c_{1}(t)=\frac{e^{4t}-4e^{2t}}{8(1+e^{2t})^{5/2}}$

and recursively

$2\sqrt{1+e^{2t}}c_{n+1}(t)=c_{n}^{\prime}(t)-\frac{e^{2t}}{1+e^{2t}}c_{n}(t)+\sum_{k=1}^{n}c_{k}(t)c_{n-k}(t).$

Plugging this into (2.12), we obtain

$\log
a(\lambda)=\nu\log(2\nu)-\nu-\frac{1}{2}\log\nu+\frac{1}{2}\log\frac{\pi}{2}+\sum_{n=1}^{\infty}\frac{\alpha_{n}-c_{n}(\nu)}{\nu^{n}}+O(|\nu|^{-\infty}),$

where

$c_{n}(\nu)=-\int_{-\log\nu}^{\infty}\sigma_{n}(t)dt.$

Moreover, it can be shown (cf. [20, Ch.10, §7]) that the coefficients
$c_{n}(\nu)$ are polynomials in $\nu/\sqrt{\nu^{2}+1}$, which gives the
asymptotic expansion of $\log a(\lambda)$.

###  3.2 Potential $q(x)=2\cosh 2x$ on the line

Here we consider the modified Mathieu differential equation

$-\psi^{\prime\prime}+2\cosh 2x\,\psi=\lambda\psi,\quad-\infty<x<\infty.$
(3.6)

It is well-known that the antiderivative of the function $\sqrt{2\cosh
2x-\lambda}$ is expressed in terms of elliptic integrals. Namely, let

$F(\varphi,k)=\int_{0}^{\varphi}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}\quad\text{and}\quad
E(\varphi,k)=\int_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}\theta}\,d\theta$

be, correspondingly, elliptic integrals of the first and second kinds, where
$0<k^{2}<1$ and $-\pi/2<\varphi<\pi/2$. Putting $\lambda=2-\nu^{2}$, where
$\nu>0$, we have for $\nu>2$ (see [23, formula (22) on p. 136])

$\displaystyle\int_{0}^{x}\sqrt{\nu^{2}-2+2\cosh 2s}\,ds$
$\displaystyle=\nu(F(\varphi,k)-E(\varphi,k))+\tanh\frac{x}{2}\,\sqrt{\nu^{2}-2+2\cosh
2x},$ (3.7)

where

$\varphi=\sin^{-1}\left(\tanh\frac{x}{2}\right)\quad\text{and}\quad
k^{2}=\frac{\nu^{2}-4}{\nu^{2}}.$ (3.8)

The corresponding solutions $\psi_{1}(x,\lambda)$ and $\psi_{2}(x,\lambda)$ of
the differential equation (3.6) with asymptotics (2.2)–(2.3) are

$\displaystyle\psi_{1}(x,\lambda)$
$\displaystyle\simeq\frac{C_{1}(\lambda)}{\sqrt[4]{2\cosh
2x+\nu^{2}}}e^{\nu(F(\varphi,k)-E(\varphi,k))+\tanh\frac{x}{2}\,\sqrt{\nu^{2}-2+2\cosh
2x}},$ (3.9) $\displaystyle\psi_{2}(x,\lambda)$
$\displaystyle\simeq\frac{C_{2}(\lambda)}{\sqrt[4]{2\cosh
2x+\nu^{2}}}e^{-\nu(F(\varphi,k)-E(\varphi,k))-\tanh\frac{x}{2}\,\sqrt{\nu^{2}-2+2\cosh
2x}},$ (3.10)

where $\lambda=2-\nu^{2}\to-\infty$ (cf. [24]).

Since $\varphi\to\pi/2$ as $x\to\infty$, we have

$\int_{0}^{x}\sqrt{\nu^{2}-2+2\cosh
2s}\,ds=\nu(\bm{K}(k)-\bm{E}(k)))+e^{x}+O(e^{-x}),$

where $\bm{K}(k)$ and $\bm{E}(k)$ are, respectively, complete elliptic
integrals of the first and second kinds. Choosing the constants in
(3.9)–(3.10) as

$C_{1}(\lambda)=\sqrt{\frac{\pi}{2}}\,e^{-\nu(\bm{K}(k)-\bm{E}(k))}\quad\text{and}\quad
C_{2}(\lambda)=\sqrt{\frac{1}{2\pi}}\,e^{\nu(\bm{K}(k)-\bm{E}(k))},$ (3.11)

we get a solution $\psi_{1}(x,\lambda)$ with the same asymptotic as
$x\to\infty$ as the modified Bessel function of the second kind
$K_{i\sqrt{\lambda}}(e^{x})$,

$\psi_{1}(x,\lambda)=\sqrt{\frac{\pi}{2e^{x}}}\,e^{-e^{x}}(1+O(e^{-x}))\quad\text{as}\quad
x\to\infty,$

and a solution $\psi_{2}(x,\lambda)$ with the same asymptotic as $x\to\infty$
as the modified Bessel function of the first kind
$I_{i\sqrt{\lambda}}(e^{x})$,

$\psi_{2}(x,\lambda)=\sqrt{\frac{1}{2\pi
e^{x}}}\,e^{e^{x}}(1+O(e^{-x}))\quad\text{as}\quad x\to\infty.$

We have $W(\psi_{1},\psi_{2})=1$ and

$\psi_{1}(x,\lambda)=t_{11}(\lambda)\psi_{1}(-x,\lambda)+t_{12}(\lambda)\psi_{2}(-x,\lambda),$
(3.12)

where

$t_{12}(\lambda)=W(\psi_{1}(x,\lambda),\psi_{1}(-x,\lambda))$ (3.13)

is an entire function of order $1/2$ with zeros — the eigenvalues
$\lambda_{n}$ of the Schrödinger operator (1.1). Corresponding Fredholm
determinant is

$a(\lambda)=\frac{t_{12}(\lambda)}{t_{12}(0)}.$

Let $\chi(x,\lambda)$ be the function defined in (2.4). Since the
antiderivative (3.7)–(3.8) is an odd function of $x$, as in Sect. 2.2.2 we get

$\displaystyle\lim_{x\to\infty}\chi(x,\lambda)$ $\displaystyle=0,$ (3.14)
$\displaystyle\lim_{x\to-\infty}\chi(x,\lambda)$ $\displaystyle=\log
a(\lambda)+\log\pi+\log t_{12}(0)-2\nu(\bm{K}(k)-\bm{E}(k)).$ (3.15)

We state the main result of this section.

######  Theorem 1.

Fredholm determinant $a(\lambda)$ of the Schrödinger operator (1.1) admits the
following asymptotic expansion as $\lambda=2-\nu^{2}\to-\infty$

$\log a(\lambda)=2\nu(\bm{K}(k)-\bm{E}(k))-\log\pi-\log
t_{12}(0)+\sum_{n=1}^{\infty}\frac{c_{n}}{\nu^{n}}+O(|\nu|^{-\infty}),$ (3.16)

where $k^{2}=1-4\nu^{-2}$ and the coefficients $c_{n}$ are determined
explicitly.

###### Proof.

It follows from (3.14)–(3.15) that

$\log a(\lambda)=2\nu(\bm{K}(k)-\bm{E}(k))-\log\pi-\log
t_{12}(0)-\int_{-\infty}^{\infty}\sigma(x,\lambda)dx,$ (3.17)

where $\sigma(x,\lambda)$ is defined in (2.5) and satisfies the Riccati
equation (2.6). Using

$2\cosh 2x-\lambda=\nu^{2}+2\cosh 2x-2=\nu^{2}+4\sinh^{2}x,$

we can rewrite the equation (2.6) as

$\displaystyle\frac{d\sigma}{dx}=-\sigma^{2}+2\sqrt{\nu^{2}+4\sinh^{2}x}\,\sigma+\frac{2\sinh
2x}{\nu^{2}+4\sinh^{2}x}\sigma$ $\displaystyle+\frac{8\cosh
2x}{\nu^{2}+4\sinh^{2}x}-5\left(\frac{\sinh
2x}{{\nu^{2}+4\sinh^{2}x}}\right)^{2}.$

It is convenient to change variables by $\sinh x=\dfrac{\nu}{2}\sinh y$, so

$\nu^{2}+4\sinh^{2}x=\nu^{2}\cosh^{2}y$

and

$\displaystyle\frac{dx}{dy}=\frac{\nu\cosh y}{2\cosh x}=\frac{\nu\cosh
y}{\sqrt{4+\nu^{2}\sinh^{2}y}}=\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!-\frac{1}{2}}.$

Next, introduce the function
$\tau(y,\lambda)=\sigma\left(\sinh^{-1}(\frac{\nu}{2}\sinh y),\lambda\right)$,
so

$\int_{-\infty}^{\infty}\sigma(x,\lambda)dx=\int_{-\infty}^{\infty}\tau(y,\lambda)\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!-\frac{1}{2}}dy.$
(3.18)

The Riccati equation for $\tau(y,\lambda)$ takes the form

$\displaystyle\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!\frac{1}{2}}\frac{d\tau}{dy}=-\tau^{2}+2\nu\cosh
y\,\tau+\tanh
y\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!\frac{1}{2}}\tau$
$\displaystyle+4\tanh^{2}y-\frac{5}{4}\tanh^{4}y+\frac{1}{\nu^{2}\cosh^{2}y}(8-\frac{5}{4}\tanh^{2}y).$
(3.19)

Now it is straightforward to show that that equation (3.19) admits an
asymptotic solution

$\tau(y,\lambda)=\sum_{n=1}^{\infty}\frac{\tau_{n}(y)}{\nu^{n}},$

where

$\tau_{1}(y)=-\frac{1}{2\cosh y}(4\tanh^{2}y-\frac{5}{4}\tanh^{4}y).$

The coefficients $\tau_{n}(y)$ are obtained recursively using the expansion

$\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!\frac{1}{2}}=\sum_{m=1}^{\infty}(-1)^{m}\binom{\frac{1}{2}}{m}\frac{k^{2m}}{\cosh^{2m}y}$

and the binomial expansion for $k^{2m}=(1-4\nu^{-2})^{m}$; only finitely many
terms from these expansions are needed in order to get the $n$-th term.
Substituting the asymptotic expansion for $\tau(y,\lambda)$ into (3.18) and
(3.17) and using

$\left(1-\frac{k^{2}}{\cosh^{2}y}\right)^{\\!\\!-\frac{1}{2}}=\sum_{m=1}^{\infty}(-1)^{m}\binom{-\frac{1}{2}}{m}\frac{k^{2m}}{\cosh^{2m}y},$

we get the desired asymptotic expansion (3.16). ∎

######  Remark.

Since

$\bm{K}(k)=\log\frac{4}{\sqrt{1-k^{2}}}+o(1)\quad\text{and}\quad\bm{E}(k)=1+o(1)\quad\text{as}\quad
k\to 1,$

we have as $\nu\to\infty$,

$\nu(\bm{K}(k)-\bm{E}(k))=\nu\log(2\nu)-2\nu+o(1).$

Here the $o(1)$ term — the remainder — is of the form
$f_{1}(\nu^{-1})+f_{2}(\nu^{-1})\log\nu$, where $f_{1}(x)$ and $f_{2}(x)$ are
convergent for $|x|<1$ power series in $x^{2}$ that vanish at $x=0$ (see [25,
p. 54]).

##  4 Existence of the solution $\psi_{1}(x,\lambda)$

Here for the convenience of the reader we prove that differential equation
(3.6) has a solution $\psi_{1}(x,\lambda)$ which asymptotically as
$x\to\infty$ behaves like the modified Bessel function of the second kind.
Namely, we have the following result.

######  Theorem 2.

Modified Mathieu equation (3.6) has a solution $\psi_{1}(x,\lambda)$ with the
following asymptotic

$\psi_{+}(x,\lambda)=K_{ik}(e^{x})(1+o(1))\quad\text{as}\quad
x\to\infty,\quad\text{where}\quad\lambda=k^{2}.$

For fixed $x$ the function $\psi_{1}(x,\lambda)$ is entire of order $1/2$.

###### Proof.

We consider the Schrödinger operator $H=-\dfrac{d^{2}}{dx^{2}}+2\cosh 2x$ as a
perturbation of the operator $H_{0}=-\dfrac{d^{2}}{dx^{2}}+e^{2x}$ by a small
as $x\to\infty$ potential $e^{-2x}$. The operator $H_{0}$ has a simple
absolutely continuous spectrum $[0,\infty)$, and for
$\lambda\in\mathbb{C}\setminus[0,\infty)$ its resolvent
$R_{\lambda}^{0}=(H_{0}-\lambda I)^{-1}$ is an integral operator in
$L^{2}({\mathbb{R}})$ with the integral kernel (see, e.g., [26, Ch. 4.15])

$R_{\lambda}^{0}(x,y)=\begin{cases}I_{-ik}(e^{x})K_{ik}(e^{y}),&\quad x\leq
y,\\\ K_{ik}(e^{x})I_{-ik}(e^{y}),&\quad x\geq y.\end{cases}$ (4.1)

Here $k=\sqrt{\lambda}$ and $\mathrm{Im}\,k>0$. The operator $H_{0}$ has a
Volterra type Green’s function

$G(x,y,k)=\begin{cases}I_{-ik}(e^{x})K_{ik}(e^{y})-K_{ik}(e^{x})I_{-ik}(e^{y}),&\quad
x\leq y,\\\ 0,&\quad x>y.\end{cases}$

Since

$K_{\nu}(z)=\frac{\pi}{2\sin\pi\nu}(I_{-\nu}(z)-I_{\nu}(z)),$

$G(x,y,k)$ is an even function of $k$ and, therefore, is an entire function of
$\lambda$ of order $1/2$.

The function $\psi_{1}(x,\lambda)$ satisfies the integral equation

$\psi_{1}(x,\lambda)=K_{ik}(e^{x})+\int_{x}^{\infty}G(x,y,k)e^{-2y}\psi_{1}(y,\lambda)dy,$
(4.2)

which can be solved by successive approximations. Indeed, put
$f_{0}(x,k)=K_{ik}(e^{x})$ and

$f_{n}(x,k)=\int_{x}^{\infty}G(x,y,k)e^{-2y}f_{n-1}(y,k)dy.$

Using the estimates

$|K_{ik}(e^{x})|\leq
Ce^{-e^{x}}e^{-\frac{1}{2}x}\quad\text{and}\quad|I_{ik}(e^{x})|\leq
Ce^{e^{x}}e^{-\frac{1}{2}x}$ (4.3)

for $a\leq x<\infty$ (the constant $C$ depends on $a$), we have
$f_{n}(x,-k)=f_{n}(x,k)$ and

$|f_{n}(x,k)|\leq\frac{2^{n}C^{2n+1}}{3^{n}n!}e^{-e^{x}}e^{-\frac{6n+1}{2}x},$
(4.4)

which can be easily proved by induction. Namely, (4.4) holds for $n=0$, and
using the estimates (4.3), we get

$\displaystyle|f_{n+1}(x,k)|\leq|I_{-ik}(e^{x})|\int_{x}^{\infty}e^{-2y}|K_{ik}(e^{y})f_{n}(y,k)|dy$
$\displaystyle+|K_{ik}(e^{x})|\int_{x}^{\infty}e^{-2y}|I_{-ik}(e^{y})f_{n}(y,k)|dy$
$\displaystyle\leq\frac{2^{n}C^{2n+3}}{3^{n}n!}\left[e^{e^{x}}e^{-\frac{1}{2}x}\int_{x}^{\infty}e^{-2y}e^{-2e^{y}}e^{-\frac{1}{2}y}e^{-\frac{6n+1}{2}y}dy\right.$
$\displaystyle\left.+e^{-e^{x}}e^{-\frac{1}{2}x}\int_{x}^{\infty}e^{e^{y}}e^{-\frac{1}{2}y}e^{-2y}e^{-e^{y}}e^{-\frac{6n+1}{2}y}dy\right]$
$\displaystyle\leq\frac{2^{n+1}C^{2n+3}}{3^{n+1}(n+1)!}e^{-e^{x}}e^{-\frac{6n+7}{2}x}.$

Thus

$\psi_{1}(x,\lambda)=\sum_{n=0}^{\infty}f_{n}(x,k)$

is given by the absolutely convergent series and

$|\psi_{1}(x,\lambda)-K_{ik}(e^{x})|\leq Ce^{-e^{x}}e^{-\frac{7}{2}x}.\qed$

As in Sect. 3.2, we have

$t_{12}(\lambda)=W(\psi_{1}(x,\lambda),\psi_{1}(-x,\lambda))=-2\psi_{1}(0,\lambda)\psi^{\prime}_{1}(0,\lambda),$

and the eigenvalue problem (3.6) is equivalent to two radial eigenvalue
problems

$\displaystyle-\psi^{\prime\prime}+2\cosh 2x\,\psi$
$\displaystyle=k^{2}\psi,\quad 0<x<\infty,$ $\displaystyle\quad\psi(0)=0,$
$\displaystyle-\psi^{\prime\prime}+2\cosh 2x\,\psi$
$\displaystyle=k^{2}\psi,\quad 0<x<\infty,$
$\displaystyle\quad\psi^{\prime}(0)=0.$

Acknowledgments. It is a pleasure to thank Ari Laptev for the discussion of
the trace identities and for drawing my attention to the reference [19]. I am
also grateful to Marcos Mariño for the useful comments and for pointing to the
references [8] and [17, 18].

## References

  * [1] M.C. Gutzwiller, The quantum mechanical Toda lattice. _Ann. Phys._ 124, 347–381 (1980); The quantum mechanical Toda lattice II., _Ann. Phys._ 133, 304–331 (1981).
  * [2] E.K. Sklyanin, The quantum Toda chain. In _Non-Linear Equations in Classical and Quantum Field Theory_ (Paris 1984), Lecture Notes in Phys. 226, Springer, Berlin,196–233 (1985).
  * [3] E.T. Whittaker, G.N. Watson, _A Course of Modern Analysis_ , Cambridge University Press, 1927.
  * [4] Q. Pasquier, M. Gaudin, The periodic Toda chain and a matrix generalization of the Bessel function recursion relations. _J. Phys._ A25, 5243–5252 (1992).
  * [5] S. Kharchev and D. Lebedev, Integral representation for the eigenfunctions of a quantum periodic Toda chain. _Lett. Math. Phys._ 50, 53–77 (1999).
  * [6] N.A. Nekrasov, S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories. In _Proceedings of the 16th International Congress on Mathematical Physics_ (Prague 2009), World Scientific, 265–289, (2010).
  * [7] M. Mariño, Spectral theory and mirror symmetry. In _String-Math 2016_ (Paris 2016), Proc. Sympos. Pure Math. 98, Amer. Math. Soc., Providence, RI, 259–294 (2018).
  * [8] A. Grassi, J. Gu and M. Mariño. Non-perturbative approaches to the quantum Seiberg-Witten curve. _J. High Energy Phys._ no. 7, 106, (2020).
  * [9] A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. _J. Funct. Anal._ (2)151, 531–545 (1997).
  * [10] A. Laptev, L. Schimmer and L. Takhtajan, Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves. _Geometric and Functional Analysis (GAFA)_ 26:1, 288–305 (2016).
  * [11] A. Laptev, L. Schimmer and L. Takhtajan, Weyl asymptotics for perturbed functional difference operators. _J. Math. Phys._ 60, 103505 (2019).
  * [12] L.A. Dikii, Trace formulas for Sturm-Liouville differential operators. _Uspekhi Mat. Nauk_ 13:3(81), 111–143 (1958) (in Russian); English transl. _Translations AMS Series 2_. 18 81–115 (1958).
  * [13] I.M. Gelfand, B.M. Levitan, On a simple identity for the eigenvalues of a second-order differential operator. _Dokl. Akad. Nauk. USSR_ 88, 953–956 (1953) (in Russian).
  * [14] V.S. Buslaev, L.D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator. _Dokl. Akad. Nauk SSSR_ 132:1, 13–16 (1960) (in Russian); English transl. _Soviet Math. Dokl._ 1, 451–454 (1960).
  * [15] V.E. Zakharov and L.D. Faddeev, Korteweg-de Vries equation: a completely integrable Hamiltonian system. _Funktsional’nyi Analiz i Ego Prilozheniya_ 5:4, 18–27 (1971) (in Russian); English transl. _Funct. Anal. Appl._ 5:4, 280–287 (1971).
  * [16] L.A. Takhtajan, Etudes of the resolvent. _Uspekhi Mat. Nauk_ 75:1(451), 155–194 (2020) (in Russian); English transl. _Russian Math. Surveys_ 75:1, 147–186 (2020).
  * [17] A. Voros, Spectre de l’équation de Schrödinger et méthode BKW. _Publications Mathématiques d’Orsay_ 81, (1982).
  * [18] A. Voros, The return of the quartic oscillator: the complex WKB method. _Ann. Inst. H. Poincaré Sect. A (N.S.)_ 39:3, 211–338 (1983).
  * [19] A. Pushnitski, I. Sorrell, High energy asymptotics and trace formulas for the perturbed harmonic oscillator. _Ann. Henri Poincaré_ 7 no. 2, 381–396 (2006).
  * [20] F. Olver, _Asymptotics and special functions_ , Academic Press, 1974.
  * [21] M.V. Fedoryuk, _Asymptotic Analysis: Linear Ordinary Differential Equations_ , Springer 1993.
  * [22] G. Pólya, Bemerkung Über die Integraldarstellung der Riemannschen $\xi$-Funktion._Acta Math._ 48:3-4, 305–317 (1926).
  * [23] I.S. Gradshtein, I.M. Ryzhik, _Tables Of Integrals, Series And Products_ , 7th Ed, Academic Press, 2007.
  * [24] N.S. Grigor’eva, _Uniform asymptotic expansions of solutions of the Mathieu equation and the modified Mathieu equation_ , Zap. Nauchn. Sem. LOMI, 62, 60–91 (1976) (in Russian); English transl. _J. Soviet Math._ 11:5, 700–721 (1979).
  * [25] A. Cayley, _An Elementary Treatise on Elliptic Functions_. George Bell and Sons, London, 1895; reprinted by Dover Publications, 1961.
  * [26] E.C. Titchmarsh, _Eigenfunction Expansions associated with Second-Order Differential Operators. Part I_. 2nd edition, Oxford University Press, London, 1962.