Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators

Generalized Hermite Polynomials
and Monodromy-Free Schrödinger Operators††This paper is a contribution to the
Special Issue on Painlevé Equations and Applications in Memory of Andrei
Kapaev. The full collection is available at
https://www.emis.de/journals/SIGMA/Kapaev.html

Victor Yu. NOVOKSHENOV

V.Yu. Novokshenov

Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Str., 450008, Ufa, Russia<EMAIL_ADDRESS>http://matem.anrb.ru/ru/novokshenovvy

Received March 20, 2018, in final form September 20, 2018; Published online
September 30, 2018

The paper gives a review of recent progress in the classification of
monodromy-free Schrödinger operators with rational potentials. We concentrate
on a class of potentials constituted by generalized Hermite polynomials. These
polynomials defined as Wronskians of classic Hermite polynomials appear in a
number of mathematical physics problems as well as in the theory of random
matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those
potentials demonstrate localized oscillatory behavior near the origin. We
derive an explicit condition of non-singularity of the corresponding
potentials and estimate a localization range with respect to indices of
polynomials and distribution of their zeros in the complex plane. It turns out
that 1D SUSY quantum non-singular potentials come as a dressing of the
harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this
end, all generalized Hermite polynomials are produced by appropriate periodic
closure of this algebra which leads to rational solutions of the Painlevé IV
equation. We discuss the structure of the discrete spectrum of Schrödinger
operators and its link to the monodromy-free condition.

generalized Hermite polynomials; monodromy-free Schrödinger operator; Painlevé
IV equation; meromorphic solutions; distribution of zeros; 1D SUSY quantum
mechanics

30D35; 30E10; 33C75; 34M35; 34M55; 34M60

In memory of my friend and colleague Andrei Kapaev

##  1 Introduction

The integrability property of Painlevé equations reveals a number of
applications of their solutions. Besides traditional self-similar modes in
nonlinear PDE’s of mathematical physics they provide new construction material
for integrable quantum mechanics and spectral theory. In this paper, we give a
brief review of recent achievements in these applications of rational
solutions of the fourth Painlevé equation (PIV). We trace how they come from
monodromy-free potentials of the Schrödinger equation and from supersymmetric
dressing of the harmonic potential in one-dimensional quantum mechanics.

Another ingredients of these interconnections are the generalized Hermite
polynomials which build all rational solutions of PIV. Their appearance in
multiple applications is explained by the determinant representation of these
polynomials. Actually, it can be set as a definition in terms of classical
Hermite polynomials. Namely, generalized Hermite polynomials (GHP)
$H_{m,n}(z)$ are defined as follows [8, 19]

$\displaystyle H_{m,n}(z)=\det\left(P_{n-i+j}(z)\right)_{i,j=1}^{m},$

where

$\displaystyle P_{s}(z)=\sum_{i+2j=s}\frac{1}{6^{j}i!j!}z^{i},$

or, equivalently, as Wronskians of classical Hermite polynomials

$\displaystyle
H_{m,n}(z)=c_{m,n}{\mathcal{W}}\left(H_{m}(z),H_{m+1}(z),\ldots,H_{{{m+n-1}}}(z)\right),$
(1.1)

where $H_{n}(z)=(-1)^{n}{\rm e}^{z^{2}}\frac{{\rm d}^{n}}{{\rm d}z^{n}}{\rm
e}^{-z^{2}}$ and $c_{m,n}$ are normalization constants.

Like classical orthogonal polynomials $H_{m,n}$ have a number of useful
properties. For example, they constitute recurrence coefficients for
orthogonal polynomials $p_{n}(x)$ with weight
$w(x,z,m)=(x-z)^{m}\exp\big{(}{-}x^{2}\big{)}$ [7, 10]

$\displaystyle xp_{n}(x)=p_{n+1}(x)+a_{n}(z,m)p_{n}(x)+b_{n}(z,m)p_{n-1}(x),$

where

$\displaystyle a_{n}(z,m)=-\frac{1}{2}\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{n+1,m}}{H_{n,m}},\qquad
b_{n}(z,m)=\frac{nH_{n+1,m}H_{n-1,m}}{2H^{2}_{n,m}}.$

Another property is a formula for rational solutions to the Painlevé IV
equation

$\displaystyle v(z)={-2z+}\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{{m,n+1}}(z)}{H_{{m+1,n}}(z)}.$ (1.2)

In this case PIV equation

$\displaystyle
v^{\prime\prime}=\frac{(v^{\prime})^{2}}{2v}+\frac{3}{2}v^{3}+4zv^{2}+2\big{(}z^{2}-a\big{)}v+{\frac{b}{v}},$
(1.3)

has integer coefficients

$\displaystyle a=n-m,\qquad b=-2(m+n+1)^{2},$

where $m$ and $n$ are the indices of the corresponding polynomials [16].

The solutions (1.2) have a specific structure of poles in the complex plane.
It can be thought of as an equilibrium state of Coulomb charged particles in
an external field. Indeed, any pole of a rational solution to PIV has residue
equal to $c_{j}=+1$ or $c_{j}=-1$ (see Theorem 3.1). The poles $z_{j}$ can be
interpreted as positive and negative charges interacting by the logarithmic
potential and influenced by the external quadratic potential

$\displaystyle
U(z_{1},z_{2},\ldots,z_{n})=\sum\limits_{j=1}^{n}c_{j}z_{j}^{2}+\sum\limits_{j\not=k}^{n}c_{j}c_{k}\log(z_{j}-z_{k})^{2}.$

The equilibrium condition provides the generalized Stiltjes relation [25]

$\displaystyle\sum\limits_{j\not=k}\frac{c_{j}}{z_{k}-z_{j}}+z_{k}=0,\qquad
k=1,2,\ldots.$

Here each pole coincides with zero of the related polynomial. The distribution
of poles for large orders of polynomials has been studied since classical
works by T. Stiltjes [24] and M. Plancherel and W. Rotach [22]. This question
was discussed recently in applications to dynamics of Coulomb log-gases [14]
and approximations by rational functions in logarithmic potential theory [23].

Since the PIV equation (1.3) is integrable in the sense of soliton theory [13,
15], all its rational solutions have been found and labeled by recursions of
Bäcklund transformations [13, Chapter 6]. These recursions can be rewritten as
dressing chains of the Lax operator with some periodic closures. As a by-
product this gives a set of Schrödinger operators $L$ formed by GHP

$\displaystyle L=-\frac{{\rm d}^{2}}{{\rm d}x^{2}}+u(x),\qquad
u(x)=f^{\prime}(x)+f^{2}(x),\qquad f(x)=-x+\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m,n}(z)}{H_{m,n+1}(z)}.$ (1.4)

These operators are monodromy-free (see Section 2) and the discrete spectrum
of each is an arithmetic sequence with a finite gap (Theorem 3.7). Moreover,
all potentials (1.4) are non-singular on the real line for odd $n$. This is
due to Theorem 3.5 below which proves $\operatorname{Res}u(z)=0$. In turn,
this follows from the distribution of zeros of GHP.

One can mention also a recent application of GHP in matrix models of
statistical physics. Consider a degenerate Gaussian unitary ensemble where
eigenvalues $\lambda_{k}$ are fixed $k=1,2,\ldots,n$, and $\lambda_{n+1}=z$
has $m$-fold multiplicity. Then the partition function of the ensemble has the
form [7]

$\displaystyle
D_{n}(z)=\frac{1}{n!}\int^{\infty}_{\infty}\cdots\int^{\infty}_{\infty}\prod_{1\leq
i<j\leq
n}(\lambda_{i}-\lambda_{j})^{2}\prod\limits_{k=1}^{n}(\lambda_{k}-z)^{m}{\rm
e}^{-\lambda_{k}^{2}}{\rm d}\lambda_{k},$ (1.5)

where

$\displaystyle D_{n}(z)=A_{m,n}H_{m,n}(cz),\qquad c={\rm
i}\sqrt{\frac{2}{3}},\qquad A_{m,n}={\rm const}.$

Note that formula (1.5) is proved with the help of dressing chains and ladder
operators discussed below in Section 4.

Figure 1: Monodromy-free potential (1.4) $u(x)$, generated by polynomials
$H_{m,n}(z)$ and $H_{m,n+1}(z)$ for $m=17$, $n=4$ (left) and zeros of
$H_{m,n}$ (blue) and $H_{m,n+1}$ (red) in the complex plane (right).

##  2 Monodromy-free potentials and dressing chain

A Schrödinger operator

$\displaystyle L=-\frac{{\rm d}^{2}}{{\rm d}z^{2}}+u(z)$ (2.1)

with meromorphic potential is called monodromy-free if all solutions of the
equation

$\displaystyle L\psi=-\psi^{\prime\prime}+u\psi=\lambda\psi$

are meromorphic in the whole complex plane $z\in\mathbb{C}$ for all $\lambda$.
In other words, monodromy of the equation (2.1) in the complex plane is
trivial for all $\lambda$.

The problem of classification of monodromy-free Schrödinger operators is
traditional in spectral theory (see [11, 18]). It became even more important
in soliton theory where monodromy-free potentials form a class of soliton
solutions to nonlinear equations for which a Schrödinger operator enters the
Lax pair. In a case of potentials decreasing at infinity on the real line the
spectral theory is well understood (see [28]). In the framework of soliton
theory there were found new classes of monodromy-free potentials such as
finite-gap ones and rational potentials with quadratic growth at infinity.
Here the latter class will be studied in detail in the special case of
rational solutions of the PIV equation.

According to [26], the Schrödinger operator (2.1) $L_{j}$ with potential
$u_{j}(z)$ is factorized in the form

$\displaystyle L_{j}=-\left(\frac{{\rm d}}{{\rm
d}z}+f_{j}(z)\right)\left(\frac{{\rm d}}{{\rm d}z}-f_{j}(z)\right),$ (2.2)

where the function $f_{j}(z)$ satisfies the Riccati equation
(${}^{\prime}={\rm d}/{\rm d}z$)

$\displaystyle f_{j}^{\prime}+f_{j}^{2}=u_{j}.$ (2.3)

The Darboux transformation

$\displaystyle L_{j}\ \mapsto\ L_{j+1}=-\left(\frac{{\rm d}}{{\rm
d}z}-f_{j}(z)\right)\left(\frac{{\rm d}}{{\rm
d}z}+f_{j}(z)\right)+\alpha_{j},$

produces the new potential

$\displaystyle
u_{j+1}=f^{\prime}_{j+1}+f^{2}_{j+1}=-f_{j}^{\prime}+f_{j}^{2}+\alpha_{j}=u_{j}-2f_{j}^{\prime}+\alpha_{j}.$

This gives rise to the dressing chain equations [26]

$\displaystyle
f^{\prime}_{j+1}+f^{\prime}_{j}=f_{j}^{2}-f_{j+1}^{2}+\alpha_{j},\qquad
j=1,2,\ldots,n,\ldots,$ (2.4)

where $\alpha_{i}$ are arbitrary constants.

In other words, equations (2.4) are equivalent to the relations between
Schrödinger operators

$\displaystyle-\left(\frac{{\rm d}}{{\rm d}z}-f_{j}(z)\right)\left(\frac{{\rm
d}}{{\rm d}z}+f_{j}(z)\right)+\alpha_{j}=-\left(\frac{{\rm d}}{{\rm
d}z}+f_{j+1}(z)\right)\left(\frac{{\rm d}}{{\rm d}z}-f_{j+1}(z)\right).$

This property plays a key role in the calculation of spectrum of the
monodromy-free potential (2.3) $u(x)=f_{1}^{\prime}(x)+f^{2}_{1}(x)$.

Following [2, 6, 26], consider the following periodic closure of the dressing
chain (2.4)

$\displaystyle f_{j}=f_{j+N},\qquad\alpha_{j}=\alpha_{j+N}.$

For $N=3$ the infinite chain (2.4) reduces to the second-order ODE written in
symmetric form (sPIV) found in [2, 6]

$\displaystyle\phi^{\prime}_{1}+\phi_{1}(\phi_{2}-\phi_{3})-\alpha_{1}=0,\qquad\phi^{\prime}_{2}+\phi_{2}(\phi_{3}-\phi_{1})-\alpha_{2}=0,$
$\displaystyle\phi^{\prime}_{3}+\phi_{3}(\phi_{1}-\phi_{2})-\alpha_{3}=0,$
(2.5)

where

$\displaystyle\phi_{1}=f_{1}+f_{2},\qquad\phi_{2}=f_{2}+f_{3},\qquad\phi_{3}=f_{3}+f_{1},$

and

$\displaystyle\phi_{1}+\phi_{2}+\phi_{3}=-2z,\qquad\alpha_{1}+\alpha_{2}+\alpha_{3}=-2.$

The system (2.5), in turn, is equivalent to the PIV equation [2]

$\displaystyle
v^{\prime\prime}=\frac{(v^{\prime})^{2}}{2v}+\frac{3}{2}v^{3}+4zv^{2}+2\big{(}z^{2}-a\big{)}v+{\frac{b}{v}},$

with

$\displaystyle v=\phi_{1},\qquad
a=\frac{1}{2}(\alpha_{{2}}-\alpha_{{3}}),\qquad
b=-\frac{1}{2}\alpha_{1}^{{2}}.$

The class of rational solutions to the system sPIV has simple “seed solutions”

$\displaystyle\phi_{1}=-\frac{1}{z},\qquad\phi_{2}=\frac{1}{z},\qquad\phi_{3}=-2z,$
(2.6)

with parameters $\alpha_{1}=\alpha_{2}={-2}$ and $\alpha_{3}={0}$. They
correspond to the “$-1/z$ hierarchy” and the “$-2z$ hierarchy” of PIV rational
solutions first found by N. Lukashevich [16]. He proved that there are no
other PIV rational solutions with leading terms $-1/z$ or $-2z$ at infinity.
Note that there is also a “$-\frac{2}{3}z$” hierarchy of PIV generated by
Okamoto polynomials [19] which we will not consider here.

The “$-1/z$” and “$-2z$” hierarchies correspond to rational solutions of the
sPIV equation (2.5) formed by generalized Hermite polynomials $H_{m,n}$ [8]

$\displaystyle\phi_{1}(z)=\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m+1,n}(z)}{H_{m,n}(z)},\qquad\phi_{2}(z)=\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m,n}(z)}{H_{m,n+1}(z)},$
$\displaystyle\phi_{3}(z)=-2z+\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m,n+1}(z)}{H_{m+1,n}(z)},$ (2.7)

where $\alpha_{1}=-2n$, $\alpha_{2}={2m+2n}$, $\alpha_{3}={-2m-2}$.

Due to the symmetry of the sPIV system (2.5) and the periodic relations
$f_{j}=f_{j+3}$ the first dressing chain component takes the form

$\displaystyle f^{(1)}_{1}(z)=-z-\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m,n}(z)}{H_{m,n+1}(z)},\qquad f^{(2)}_{1}(z)=z+\frac{{\rm
d}}{{\rm d}z}\ln\frac{H_{m+1,n}(z)}{H_{m,n+1}(z)},$ $\displaystyle
f^{(3)}_{1}(z)=-z{+}\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m,n}(z)}{H_{m+1,n}(z)}.$ (2.8)

Note that formulas (2.8) can be derived also from results of A. Oblomkov [21].
He proved that any monodromy-free potential of a Schrödinger operator (2.2)
with

$\displaystyle f_{j}=\sum\limits_{k=1}^{N}\frac{c_{k}}{z-z_{k}}-z$

is quadratic at infinity and has the form

$\displaystyle u(z)=z^{2}-2\frac{{\rm d}^{2}}{{\rm
d}z^{2}}\ln\mathcal{W}(H_{m}(z),H_{m+1}(z),\ldots,H_{m+n{-1}}(z)),$ (2.9)

where $\mathcal{W}$ is the Wronskian and $H_{k}$ are classical Hermite
polynomials. This form of the potential can easily be derived from the
definition $u=f_{1}^{\prime}+f_{1}^{2}$ by using relations (2.8)

$\displaystyle u(z)=z^{2}-2\frac{{\rm d}^{2}}{{\rm d}z^{2}}\ln
H_{m+1,n}(z)+2n-1,$ (2.10a) $\displaystyle u(z)=z^{2}-2\frac{{\rm d}^{2}}{{\rm
d}z^{2}}\ln H_{m,n+1}(z)+2n-2m+1,$ (2.10b) $\displaystyle
u(z)=z^{2}-2\frac{{\rm d}^{2}}{{\rm d}z^{2}}\ln H_{m,n}(z)-2m-1.$ (2.10c)

Taking into account the definition (1.1) we come to the representation (2.9)
up to a constant term.

##  3 Non-singular potentials on the real line

Spectral theory of Schrödinger operators (2.2) usually supposes a potential
$u(x)$ being non-singular, i.e., belonging to some functional space like
$L_{2}(\mathbb{R})$. This is true especially for applications like quantum
mechanics as we will discuss in Section 4.

We begin with the structure of poles of rational PIV solutions (2.8). In
general, all solutions of (1.3) are meromorphic functions. They are described
by the following

######  Theorem 3.1 ([25]).

Any rational solution to equation (1.3) has the form

$\displaystyle v(z)=\epsilon
z+\sum\limits_{j}\frac{c_{j}}{z-z_{j}},\qquad\epsilon=0,-\frac{2}{3},-2,\qquad
c_{j}=\pm 1,\qquad j=1,2,\ldots,$ (3.1)

and the generalized Stieltjes relation is true

$\displaystyle\sum\limits_{j\not=k}\frac{c_{j}}{z_{k}-z_{j}}+(\epsilon+1)z_{k}=0,\qquad
k=1,2,\ldots.$ (3.2)

###### Proof.

Take a Laurent series near a pole $z=z_{j}$

$\displaystyle v(z)=\sum\limits_{k=-l}^{\infty}c_{k}(z-z_{j})^{k},\qquad z\to
z_{j}.$ (3.3)

Balancing the leading terms of the series in equation (1.3) yield $l=1$,
$c_{-1}^{2}=1$. A similar comparison at infinity proves that rational solution
$u(z)$ has at most linear growth $u(z)=\epsilon z+O(1)$ as $z\to\infty$ with
$\epsilon=0,-2/3,-2$. Expand a rational solution $u(z)$ into simple fractions
(3.1) and put it into equation (1.3) looking for terms of order
$O(z-z_{k})^{-2}$ as $z\to z_{k}$. Balancing these terms gives the relations
(3.2) for any pole $z_{k}$. ∎

######  Corollary 3.2.

For any solution $v$ of the PIV equation (1.3) the residues of the function
$(z+v(z))^{2}$ are zero at any pole $z=z_{k}$

$\displaystyle\operatorname{Res}(z+v(z))^{2}=0.$

###### Proof.

From the Laurent series (3.3) one easily derives $c_{0}=-z_{j}$. This yields
the similar asymptotics for $z+v(z)$

$\displaystyle
z+v(z)=\frac{c_{-1}}{z-z_{j}}+(c_{0}+z_{j})+(c_{1}+1)(z-z_{j})+\cdots$
$\displaystyle\hphantom{z+v(z)}{}=\frac{c_{-1}}{z-z_{j}}+(c_{1}+1)(z-z_{j})+\cdots.$
∎

Each pole of the functions (2.8) comes from a zero of a GHP $H_{m,n}$,
$H_{m,n+1}$ or $H_{m+1,n}$. The polynomials satisfy the recurrence relations
[8]

$\displaystyle
2mH_{m+1,n}H_{m-1,n}=H_{m,n}H^{\prime\prime}_{m,n}-(H^{\prime}_{m,n})^{2}+2mH^{2}_{m,n},$
$\displaystyle
2nH_{m,n+1}H_{m,n-1}=-H_{m,n}H^{\prime\prime}_{m,n}+(H^{\prime}_{m,n})^{2}+2nH^{2}_{m,n},$
(3.4)

with initial conditions

$\displaystyle H_{0,0}=H_{0,1}=H_{1,0}=1,\qquad H_{1,1}=2z.$

One can easily prove by induction that solutions of the system (3.4) are
polynomials and each $H_{m,n}$ has $mn$ simple zeros. Due to the obvious
symmetries

$\displaystyle H_{m,n}(-z)=(-1)^{mn}H_{m,n}(z),\qquad H_{m,n}({\rm i}z)={\rm
i}^{mn}H_{{n,m}}(z),$

all zeros form a symmetric pattern with respect to real and imaginary axes.

Note that $H_{m,1}=H_{m}$, where $H_{m}$ is the classical Hermite polynomial
of $m$-th order, $H_{m}(z)=(-1)^{m}{\rm e}^{z^{2}}\frac{{\rm d}^{m}}{{\rm
d}z^{m}}{\rm e}^{-z^{2}}$. This means that all zeros of $H_{m,1}$ are on the
real line. However, all polynomials $H_{m,n}$ with even $n$ do not have any
real-valued zeroes. This follows from the theorem of V.E. Adler

######  Theorem 3.3 ([1]).

For $x\in\mathbb{R}$ all Wronskians
${\mathcal{W}}(H_{m_{1}}(x),H_{m_{2}}(x),\ldots,H_{m_{n}}(x))\not=0$ if
$m_{1}<m_{2}<\dots<m_{n}$ and $n$ is even.

The typical pattern of zeros is a slightly deformed rectangular region shown
in Fig. 1 (right). Its horizontal and vertical ranges are proportional to
$\sqrt{2m+n}$ and $\sqrt{2n+m}$ respectively. Recently, the generalized
Hermite polynomials have been studied in the limit $m,n\to\infty$ in a number
of papers [5, 12, 17, 20]. In the paper [17], the distribution of zeros of
$H_{m,n}(z)$ was obtained in the asymptotic limit $m\to\infty$, $n=O(1)$. On
the other hand, the paper [5] contains an analysis of $H_{m,n}(z)$ in the
limit $m,n\to\infty$, $m=rn$, $r=O(1)$ and deduces in particular bounds for
the deformed rectangular region containing the zeros. The asymptotic
distribution of zeros in the latter asymptotic regime, i.e., the
generalization of Plancherel–Rotach formulas to $H_{m,n}(z)$ for $z$ within
the deformed rectangular region for $m$ and $n$ of similar large order,
remains an open problem. We note that both papers [5, 17] apply methods of
asymptotic “undressing” of Riemann–Hilbert problems.

######  Remark 3.4.

Note that in [5] a mistake was corrected in the asymptotics found in [20].
Namely, there was an incorrect leading term of the Riemann surface equation
which led to genus-0 functions instead of genus-1 functions responsible for
the asymptotic distribution of zeros. In turn, the poles near each vertex of
the asymptotic “rectangle” (see Fig. 1 right) were found incorrectly.

We now prove that the potentials $u(z)$ provided by (2.8) are non-singular on
the real line.

######  Theorem 3.5.

The potential $u(x)=f^{\prime}_{1}(x)+f_{1}^{2}(x)$ is non-singular at
$x\in\mathbb{R}$ if

$\displaystyle f_{1}=f^{(1)}_{1},\qquad n=2k,$ (3.5a) $\displaystyle
f_{1}=f^{(2)}_{1},\qquad n=2k+1,$ (3.5b) $\displaystyle
f_{1}=f^{(3)}_{1},\qquad n=2k,$ (3.5c)

and $m$ is arbitrary.

###### Proof.

Note that the functions (2.8) $f_{1}$ can be written as

$\displaystyle f_{1}(z)=-z-v(z),$

with $v=\phi_{j+1}$ as $f_{1}=f_{1}^{(j)}$, $j=1,2,3$, $\phi_{4}=\phi_{1}$.

Then, by Corollary 3.2,

$\displaystyle
f_{1}^{2}=(z+v(z))^{2}=\frac{c^{2}_{-1}}{(z-z_{j})^{2}}+O(1),\qquad z\to
z_{j}$

for every pole $z=z_{j}$ of $v$. This yields

$\displaystyle
u=f_{1}^{\prime}+f_{1}^{2}=\frac{-c_{-1}+c^{2}_{-1}}{(z-z_{j})^{2}}+O(1),\qquad
z\to z_{j}.$

By Theorem 3.1, $c_{-1}=\pm 1$, and the pole of $u$ vanishes only for
$c_{-1}=1$. Since all GHP have simple zeros,
$H_{i,j}(z)=a(z-z_{1})\cdots(z-z_{ij})$, we have

$\displaystyle c_{-1}=\mathop{\rm Res}\limits_{z=z_{j}}\frac{{\rm d}}{{\rm
d}z}\ln\frac{H_{m+1,n}(z)}{H_{m,n+1}(z)}=\begin{cases}\hphantom{-}1,&H_{m+1,n}(z_{j})=0,\\\
-1,&H_{m,n+1}(z_{j})=0,\end{cases}$

Thus one should eliminate real-valued zeros $z_{j}\in\mathbb{R}$ that provide
$c_{-1}=-1$, i.e., the real-valued zeros of denominators in (2.7).

As for the cases (3.5a) and (3.5c), the denominators in (2.7) are $H_{m,n}$
and $H_{m+1,n}$ respectively. If $n=2k$ those polynomials do not have real-
valued zeros due to Theorem 3.3. The real-valued zeros of $H_{m,n+1}$ in this
case provide residues $c_{-1}=1$, so that $u=f_{1}^{\prime}+f_{1}^{2}$ is non-
singular.

In the case (3.5b) we have $n=2k+1$ which yields the denominator $H_{m,n+1}$
to have no zeros on the real line. Here the zeros of the numerator provide
residues $c_{-1}=1$, and $u=f_{1}^{\prime}+f_{1}^{2}$ again is non-singular. ∎

######  Remark 3.6.

The statement of Theorem 3.5 is in line with the representation (2.9) of the
potential $u(x)$ by A. Oblomkov [21]. Due to Theorem 3.3 the logarithmic
derivatives in (2.10) are non-singular if $n$ is even in cases a) and c) and
$n$ is odd in case b).

We now discuss the spectrum of Schrödinger operators (2.2) with non-singular
potentials (2.3)

$\displaystyle L\psi=\lambda\psi,\qquad L=-\left(\frac{{\rm d}}{{\rm
d}x}+f\right)\left(\frac{{\rm d}}{{\rm d}x}-f\right)=-\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+u(x).$

All potentials $u$ are quadratic at infinity. Note that the quadratic
potential itself $u(x)=x^{2}-1$ has discrete spectrum which is the set of even
numbers, $\operatorname{Sp}\big{(}{-}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+x^{2}-1\big{)}=\\{\lambda_{k}=2k,\,k=0,1,\ldots\\}$. Actually, the
GHP potentials demonstrate similar features.

######  Theorem 3.7.

If the operator $L$ has potentials (2.3) $u=f^{\prime}+f^{2}$ with
$f=f_{1}^{(j)}$, $j=1,2,3$ in (2.8), then its spectrum is discrete and
consists of even numbers with the exception or addition of a finite number of
terms

$\displaystyle\operatorname{Sp}(L)=\\{\lambda=2k,\,k=0,1,\ldots,m,m+n+1,\ldots\\},$
(3.6a)
$\displaystyle\operatorname{Sp}(L)=\\{\lambda=2k,\,k=-m,-m+1,\ldots,0,n+1,n+2,\ldots\\},$
(3.6b)
$\displaystyle\operatorname{Sp}(L)=\\{\lambda=2k,\,k=-m-n,-m-n+1,\ldots,-n-1,0,1,\ldots\\}.$
(3.6c)

###### Proof.

Since all potentials (2.3) came from iterations of Darboux transformations by
the dressing chain (2.4), their spectra are computed explicitly by M. Crum’s
method [9]. Namely, if the potential $u_{1}$ has discrete spectrum with
eigenfunctions $\psi_{k}$ corresponding to distinct $\lambda_{k}$,
$k=1,2,\ldots$, then $n$-th Darboux iteration $u_{n}$ has the form

$\displaystyle u_{n}(z)=u_{1}(z)-2\frac{{\rm d}^{2}}{{\rm
d}z^{2}}\ln\mathcal{W}(\psi_{1}(z),\ldots,\psi_{n}(z)),$

and its eigenfunctions are

$\displaystyle\psi_{n,k}=\frac{\mathcal{W}(\psi_{1},\ldots,\psi_{n},\psi_{k})}{\mathcal{W}(\psi_{1},\ldots,\psi_{n})},\qquad
k>n.$ (3.7)

Here $n$ should be even, else the denominator in (3.7) has real-valued zeroes
and $\psi_{n,k}$ have non-integrable singularities. Since
$\psi_{1},\ldots,\psi_{n}$ are no longer eigenfunctions of new potential
$u_{n}$, the spectrum of $u_{n}$ coincides with
$\operatorname{Sp}\big{(}{-}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+u_{1}\big{)}\setminus\\{\lambda_{1},\ldots,\lambda_{n}\\}$.

In particular, put $u_{1}=x^{2}-1$ and $\psi_{1}(x)={\rm
e}^{-x^{2}/2}H_{m+1}(x),\ldots,\psi_{n}(x)={\rm e}^{-x^{2}/2}H_{m+n}(x)$,
where $H_{k}$ are classical Hermite polynomials. Then

$\displaystyle u_{n}(z)=z^{2}-2\frac{{\rm d}^{2}}{{\rm
d}z^{2}}\ln\mathcal{W}(H_{m+1}(z),\ldots,H_{m+n})+2n-1$
$\displaystyle\hphantom{u_{n}(z)}{}=z^{2}-2\frac{{\rm d}^{2}}{{\rm d}z^{2}}\ln
H_{m+1,n}(z)+2n-1,$

which follows from the representation (2.10a). Since
$\operatorname{Sp}\big{(}{-}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+x^{2}-1\big{)}=2\mathbb{N}$ and $\lambda_{k}=2(m+k)$, we come to
formula (3.6a).

The other two cases (3.6b) and (3.6c) are proved in a similar way. The
spectral gap for the potential (2.10b) is $\\{2m,2(m+n)\\}$ and the constant
shift is $-2m$ with respect to (2.10a). As for (2.10c), the gap is
$\\{2m,2(m+n-1)\\}$ and the constant shift is $-2m-2n$. Applying Crum’s
formulas this gives spectra (3.6b) and (3.6c). ∎

Note that the case $f=f_{1}^{(1)}$ and the corresponding spectrum (3.6a) was
first found in the paper [1] by V.E. Adler. In the next section we reproduce
Theorem 3.7 by the Darboux dressing procedure of the harmonic potential.

##  4 1D SUSY quantum mechanics and the PIV equation

The idea to factorize quantum Hamiltonians and get supersymmetric potentials
dates back to the pioneering paper by E. Witten [27]. Later came examples of
one-dimensional realizations of this idea with the simplest polynomial
Heisenberg algebras. A connection between the harmonic oscillator and
supersymmetric (SUSY) partner potentials generated by this algebra has been
long known. Recently these potentials were identified with rational solutions
of the PIV equation. Here we follow the papers by D. Bermudes and D.J.
Fernández C. [3, 4] describing this application.

Starting with two Schrödinger operators (2.1), say $L_{j}$ and $L_{j+1}$, one
can factorize them as shown in Section 2

$\displaystyle L_{j}=A_{j}^{+}A_{j}^{-}+\epsilon_{j},\qquad
L_{j+1}=A_{j}^{-}A_{j}^{+}+\epsilon_{j},$

where

$\displaystyle A_{j}^{+}=\frac{{\rm d}}{{\rm d}x}+f_{j}(x),\qquad
A_{j}^{-}=\frac{{\rm d}}{{\rm d}x}-f_{j}(x)$ (4.1)

and $u_{j}\mapsto u_{j}+\epsilon_{j}$,
$\alpha_{j}=2(\epsilon_{j}-\epsilon_{j+1})$ in (2.3), (2.4).

Then an intertwining relation holds, namely, $L_{j+1}A_{j}^{+}=A_{j}^{+}L_{j}$
which generates the $k$-th order intertwining operators

$\displaystyle L_{k}B_{k}^{+}=B_{k}^{+}L_{1},\qquad
L_{1}B_{k}^{-}=B_{k}^{-}L_{k},\qquad B_{k}^{+}=A_{k}^{+}\cdots
A_{1}^{+},\qquad B_{k}^{-}=A_{1}^{-}\cdots A_{k}^{-}.$ (4.2)

This represents the standard SUSY algebra

$\displaystyle[\mathcal{Q}_{a},\mathcal{H}]=0,\qquad\\{\mathcal{Q}_{a},\mathcal{Q}_{b}\\}=\delta_{ab}\mathcal{H},\qquad
a,b=1,2,$
$\displaystyle\mathcal{Q}_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&B_{k}^{+}\\\
B_{k}^{-}&0\end{pmatrix},\qquad\mathcal{Q}_{2}=\frac{1}{i\sqrt{2}}\begin{pmatrix}0&B_{k}^{+}\\\
-B_{k}^{-}&0\end{pmatrix},\qquad\mathcal{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}B_{k}^{+}B_{k}^{-}&0\\\
0&B_{k}^{-}B_{k}^{+}\end{pmatrix}.$

where $\\{\,,\\}$ is the anticommutator and $\mathcal{H}$ is the Hamiltonian
with superpotential partner $u_{k}=u_{1}+2(f_{1}+\dots+f_{k})^{\prime}$ of the
initial potential $u_{1}$ [3].

Similarly, the polynomial Heisenberg algebra for the Hamiltonian (2.1) is
formed by the dressing chain operators

$\displaystyle\big{[}L,B_{k}^{\pm}\big{]}=\pm
B_{k}^{\pm},\qquad\big{[}B_{k}^{-},B_{k}^{+}\big{]}=N_{k}(L+I)-N_{k}(L),\qquad
N_{k}(L)=\prod_{j=1}^{k}(L-\epsilon_{j}).$

For $k=3$ this algebra can produce new solutions of the PIV equation starting
from the known ones (see [4]). Taking a closure condition $L_{4}=L_{1}-I$ we
come to the dressing chain (2.5) with some $\alpha_{1}$, $\alpha_{2}$ and
$\alpha_{3}$. An example is given at the end of this section.

First we start with the trivial solution of the “$-2x$ hierarchy” (2.6), which
corresponds to the harmonic potential $u_{0}(x)=x^{2}{-1}$. The eigenfunctions
of the Schrödinger operator with the harmonic potential

$\displaystyle L\psi_{0}={2}\epsilon\psi_{0}$

are written explicitly

$\displaystyle\psi_{0}(x)=\sqrt{\pi}\big{(}(1-c+{\rm i}(1+c){\rm e}^{{\rm
i}\pi\epsilon}\big{)}D_{\epsilon-1}\big{(}\sqrt{2}x\big{)}-2(1+c){\rm e}^{{\rm
i}\pi\epsilon/2}\sin(\pi\epsilon)\Gamma(\epsilon)D_{-\epsilon}\big{(}{\rm
i}\sqrt{2}x\big{)},\\!\\!\\!$ (4.3)

where $D_{\nu}(x)$ is the Weber–Hermite function, $\Gamma$ is the gamma
function and $c$ is an arbitrary complex-valued constant.

Apply now the ladder operators (4.2) to the basic eigenfunction (4.3). The
first SUSY partner potential for $k=1$ to $u_{0}(x)=x^{2}{-1}$ has the form

$\displaystyle u_{1}(x)=x^{2}-{2}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}\ln\psi_{0}(x){-1}.$

Similarly, the $k$-th potential becomes [3]

$\displaystyle u_{k}(x)=x^{2}-{2}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}\ln\mathcal{W}(\psi_{01},\psi_{02},\ldots,\psi_{0{k}}){-1},$ (4.4)

where $\mathcal{W}$ is the Wronskian of eigenfunctions (4.3) $\psi_{0j}(x)$
with parameters $\epsilon=\epsilon_{j}$.

This gives a way to construct a set of rational potentials associated with
GHP. Take parameters $\epsilon_{j}$ to be integers, because in this case the
Weber–Hermite functions in (4.3) become Hermite polynomials multiplied by the
exponential ${\rm e}^{-x^{2}/2}$. Namely, if we put

$\displaystyle\epsilon_{1}=m+1,\qquad\epsilon_{j}=m+j,\qquad c_{j}=-1,\qquad
j=2,3,\ldots,n,\qquad m,n\in\mathbb{N},$

the second term in formula (4.3) vanishes and the potential (4.4) takes the
form

$\displaystyle u_{n}(x)=x^{2}-2\frac{{\rm d}^{2}}{{\rm
d}x^{2}}\ln\mathcal{W}\left(H_{m}(x),H_{m+1}(x),\ldots,H_{m+n}(x)\right)+2n-1$
$\displaystyle\hphantom{u_{n}(x)}{}=x^{2}-2\frac{{\rm d}^{2}}{{\rm d}x^{2}}\ln
H_{m,n+1}(x)+2n-1.$

Unfortunately, this choice of $\psi_{0}$ leads to singularities in the
intermediate eigenfuntions (3.7) of the potential (4.4)

$\displaystyle\psi_{k,j}=\frac{\mathcal{W}(\psi_{01},\psi_{02},\ldots,\psi_{0j-1})}{\mathcal{W}\left(\psi_{01},\psi_{02},\ldots,\psi_{0j}\right)},\qquad
j=2,3,\ldots,k.$ (4.5)

To avoid this and make the formal dressing procedure correct, we use the
following theorem proved by D. Bermudez and D.J. Fernández C. in [4] providing
non-singularity of eigenfunctions.

######  Theorem 4.1.

Schrödinger operators (2.1) with potentials (4.4) are monodromy-free. All
potentials (4.4) and eigenfunctions (4.5) are non-singular for
$x\in\mathbb{R}$ if the dressing parameters satisfy the conditions

$\displaystyle\epsilon_{k}<\epsilon_{k-1}<\dots<\epsilon_{1}<0,$ (4.6)
$\displaystyle|c_{2j}|<1,\qquad|c_{2j+1}|>1,\qquad j=0,1,\ldots,k,$ (4.7)

where $\epsilon_{j}=\epsilon$ and $c_{j}=c$ in (4.3).

###### Proof.

Since the Weber–Hermite functions $D_{\nu}(z)$ in (4.3) are entire functions,
so is the Wronskian (4.4). Thus its logarithmic derivative is a meromorphic
function. Moreover, any solution $\psi(z,\lambda)$ of the Schrödinger equation
with potential (4.4) can be found via a finite dressing chain of $k$ operators
$A_{j}^{\pm}$ (4.1), i.e., a finite number of Darboux transformations. This
yields trivial monodromy of $\psi(z,\lambda)$.

It is easy to check that the function (4.3) has no real-valued zeros if
$\epsilon<0$ and $|c|<1$. The remaining inequalities (4.6) and (4.7) are
proved by induction (see [3] for details). ∎

Apply now Theorem 4.1 to get non-singular functions (4.5). Take parameters
$\epsilon_{j}$ and $c_{j}$ in the form

$\displaystyle\epsilon_{1}=-n,\\!\\!\qquad\epsilon_{j}=-n-j,\\!\\!\qquad
c_{2j}=0,\\!\\!\qquad c_{2j-1}=\infty,\\!\\!\qquad
j=1,2,\ldots,m-1,\\!\\!\qquad m,n\in\mathbb{N}.$

In the case of odd $j$ the eigenfunctions (4.3) are normalized as
$\psi_{0j}=D_{-\epsilon}\big{(}{\rm i}\sqrt{2}x\big{)}$. This forces the first
term in (4.3) to be zero while the second term turns into the Hermite
polynomial ${\rm e}^{x^{2}/2}H_{n+j}({\rm i}x)$. Thus the potential (4.4)
becomes

$\displaystyle u_{n}(x)=x^{2}-2\frac{{\rm d}^{2}}{{\rm
d}x^{2}}\ln\mathcal{W}(H_{n}({\rm i}x),H_{n+1}({\rm i}x),\ldots,H_{n+m-1}({\rm
i}x))-2m-1$ $\displaystyle\hphantom{u_{n}(x)}{}=x^{2}-2\frac{{\rm d}^{2}}{{\rm
d}x^{2}}\ln H_{m,n}(x)-2m-1,$ (4.8)

because $H_{k,j}({\rm i}x)={\rm i}^{kj}H_{j,k}(x)$.

As shown in [4], the spectrum of the potential (4.8) is

$\displaystyle\operatorname{Sp}\left(-\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+u_{n}\right)=\\{2\epsilon_{m-1}-2,\ldots,2\epsilon_{1}-2\\}\cup\\{0,2,4,\ldots\\},$

which coincides with formula (2.10c) found in Theorem 3.7.

Finally we illustrate the dressing of the “$-2x$ hierarchy” by the ladder
operators (4.1), (4.2) with singular eigenfunctions for the case $k=3$

$\displaystyle B_{3}^{\pm}=\left(\frac{{\rm d}}{{\rm d}x}\pm
f_{1}\right)\left(\frac{{\rm d}}{{\rm d}x}\pm f_{2}\right)\left(\frac{{\rm
d}}{{\rm d}x}\pm f_{3}\right).$ (4.9)

The closure condition $L_{4}=L_{1}-I$ leads to the system

$\displaystyle
f_{1}^{\prime}+f_{2}^{\prime}=f_{1}^{2}-f_{2}^{2}-2(\epsilon_{1}-\epsilon_{2}),$
$\displaystyle
f_{2}^{\prime}+f_{3}^{\prime}=f_{2}^{2}-f_{3}^{2}-2(\epsilon_{2}-\epsilon_{3}),$
$\displaystyle
f_{3}^{\prime}+f_{1}^{\prime}=f_{3}^{2}-f_{1}^{2}-2(\epsilon_{3}-\epsilon_{1}{-1}),$

which is equivalent to the sPIV system (2.5), where

$\displaystyle\alpha_{1}=2(\epsilon_{1}-\epsilon_{2}),\qquad\alpha_{2}=2(\epsilon_{2}-\epsilon_{3}),\qquad\alpha_{3}=2(\epsilon_{3}-\epsilon_{1}{-1}).$

Figure 2: Non-singular potential $u(x)=f_{2}^{\prime}(x)+f^{2}_{2}(x)$
generated by ladder operator (4.9) (left) and its singular component (4.10)
$f_{2}(x)$.

Choosing integer values of $\epsilon_{1}$, $\epsilon_{2}$ and $\epsilon_{3}$
it is easy to reproduce the generalized Hermite polynomials which form
rational solutions of PIV (2.7). For example, the choice

$\displaystyle{\epsilon_{3}=-3,\qquad\epsilon_{1}=1,\qquad\epsilon_{2}=4,}$

yields $m=4$ and $n=3$. This provides a rational solution of PIV in the form
(2.7)

$\displaystyle\phi_{2}(x)=\frac{{\rm d}}{{\rm
d}x}\ln\frac{H_{4,3}(x)}{H_{4,4}(x)}=\frac{64x^{3}\big{(}7875-900x^{4}+720x^{8}+64x^{1}2\big{)}}{23625+16x^{4}\big{(}7875-450x^{4}+16x^{8}\big{(}15+x^{4}\big{)}\big{)}}$
$\displaystyle\hphantom{\phi_{2}(x)=}{}+\frac{24x\big{(}{-}225+2x^{2}\big{(}{-}75-60x^{2}+120x^{4}-40x^{6}+16x^{8}\big{)}\big{)}}{675+4x^{2}\big{(}{-}675-225x^{2}+4x^{4}\big{(}{-}30+45x^{2}-12x^{4}+4x^{6}\big{)}\big{)}}.$

Since $n$ is odd, by Theorem 3.5 one should choose a non-singular potential
from the function (2.8), (3.5b)

$\displaystyle f_{2}(x)=f_{1}^{(2)}=x+\frac{{\rm d}}{{\rm
d}x}\ln\frac{H_{5,3}(x)}{H_{4,4}(x)}$
$\displaystyle=x+\frac{1}{x}-\frac{64x^{3}\big{(}7875-900x^{4}+720x^{8}+64x^{12}\big{)}}{23625+16x^{4}\big{(}7875+2x^{4}\big{(}{-}225+8x^{4}\big{(}15+x^{4}\big{)}\big{)}\big{)}}$
(4.10)
$\displaystyle\hphantom{=}{}+\frac{4x\big{(}{-}7875+4x^{2}\big{(}{-}4725+2025x^{2}-4200x^{4}+2700x^{6}-720x^{8}+112x^{10}\big{)}\big{)}}{23625+2x^{2}\big{(}{-}7875+2x^{2}\big{(}{-}4725+2x^{2}\big{(}675+2x^{2}\big{(}{-}525+270x^{2}-60x^{4}+8x^{6}\big{)}\big{)}\big{)}\big{)}}.$

The non-singular potential has the form

$\displaystyle u(x)=f_{2}^{\prime}(x)+f_{2}^{2}(x)=x^{2}-1+R(x),$

where $R$ is rational function, $R(x)=O\big{(}x^{-2}\big{)}$, $x\to\infty$.
The functions $u$ and $f_{2}$ are plotted in Fig. 2. The spectrum of the
Schrödinger operator $L$ with this potential is

$\displaystyle\operatorname{Sp}(L)=\\{-8,-6,-4,-2\\}\cup\\{8,10,12,\dots\\}.$

### Acknowledgements

The work has been supported by Russian Scientific Foundation grant
17-11-01004. The author also is grateful to the referee remarks which helped
to improve the paper.

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