# Spectral Theory of Exceptional Hermite Polynomials

David Gómez-Ullate  Escuela Superior de Ingeniería, U. Cádiz, 11519 Puerto
Real, Spain, and Departamento de Física Teórica, U. Complutense, 28040 Madrid,
Spain<EMAIL_ADDRESS>,  Yves Grandati  L. Physique et Chimie
Théoriques, U. de Lorraine, 57078 Metz, Cedex 3, France  yves.grandati@univ-
lorraine.fr and  Robert Milson  Dept. of Mathematics and Statistics, Dalhousie
U., Halifax NS, Canada B3H 3J5<EMAIL_ADDRESS>
###### Abstract.

In this paper we revisit exceptional Hermite polynomials from the point of
view of spectral theory, following the work initiated by Lance Littlejohn.
Adapting a result of Deift, we provide an alternative proof of the
completeness of these polynomial families. In addition, using equivalence of
Hermite Wronskians we characterize the possible gap sets for the class of
exceptional Hermite polynomials.

##  1\. Introduction

Consider a Sturm-Liouville problem (SLP) on $(-\infty,\infty)$:

(1) $-(Wy^{\prime})^{\prime}-Ry=\varepsilon Wy,$

where $W,R=O(e^{-x^{2}})$ as $|x|\to\infty$. Remarkably, there exist a large
class of such eigenvalue problems with polynomial eigenfunctions. Classical
Hermite polynomials, which correspond to the case of $W=e^{-x^{2}},R=0$, are
just one, very particular, example. Families of polynomials that arise as
eigenfunctions of such SLP are called exceptional Hermite orthogonal
polynomials.

Being the eigenfunctions of an SLP, exceptional Hermite polynomials are
complete families of orthogonal polynomials with weight
$W(x)>0,\;x\in(-\infty,\infty)$ [10, 8]. However, unlike the classical Hermite
polynomials, the degree sequence of an exceptional family has a finite number
of gaps in the degree sequence; there is a finite number of so-called
exceptional degrees for which there is no corresponding eigenpolynomial.

The well-known Bochner theorem [1] asserts that if $y_{n}(x),\;n=0,1,2,\ldots$
with $\deg y_{n}=n$ is a family of polynomials that satisfies a second-order
eigenvalue equation

$py_{n}^{\prime\prime}+qy_{n}^{\prime}+ry_{n}=\varepsilon_{n}y,$

then necessarily $p(x),q(x)$ are polynomials with $\deg p\leq 2,\deg q\leq 1$
and $r$ is a constant. Multiplying (1) by $-W^{-1}$ gives

(2) $y^{\prime\prime}+qy^{\prime}+ry=-\varepsilon y$

where

$q=W^{-1}W^{\prime},\quad r=W^{-1}R.$

In order for (2) to have infinitely many polynomial eigenfunctions, it is
necessary for $q(x),r(x)$ to be rational functions with $\deg q\leq 1$ and
$\deg r\leq 0$. To obtain non-classical polynomials, it is necessary for $q$
and $r$ to have poles. Bochner’s theorem then implies that such a non-
classical family of polynomial eigenfunctions must have gaps in the degree
sequence.

If we assume that there is at most a finite number of such gaps, then it is
possible to show (see [6, 8]) that, necessarily, (2) takes the form

(3)
$y^{\prime\prime}-2\left(x+\frac{\eta^{\prime}}{\eta}\right)y^{\prime}+\left(\frac{\eta^{\prime\prime}}{\eta}+2x\frac{\eta^{\prime}}{\eta}\right)y=\varepsilon
y,$

where $\eta(x)$ is a real polynomial without any real zeros . Not every choice
of $\eta$ results in an eigenvalue relation (3) with polynomial
eigenfunctions. When it does, however, one can show that there are precisely
$\deg\eta$ “exceptional” degrees.

Some examples of exceptional polynomials were investigated back in the early
90s [4] but their systematic study started about 10 years ago, where a full
classification was given for codimension one [10]. The role of Darboux
transformations in the construction process was quickly recognized [11, 25,
28] and the next conceptual step involved the generation of exceptional
families by multiple-step or higher order Darboux transformations [12].

Exceptional polynomials appear in mathematical physics as bound states of
exactly solvable rational extensions [8, 23, 25] and exact solutions to
Dirac’s equation [26]. They appear also in connection with super-integrable
systems [24, 22] and finite-gap potentials [16]. From a mathematical point of
view, the main results are concerned with the full classification of
exceptional polynomials [6], properties of their zeros [14, 17, 18], and
recurrence relations [5, 13, 19].

Lance Littlejohn and collaborators wrote a series of papers analyzing the
spectral-theoretic properties of 1-step exceptional operators [20, 21]. It is
the ambition of the present work to extend this type of analysis to the class
of multi-step exceptional Hermite operators. There are two primary results.
First, we merge the approach of Littlejohn with spectral-theoretic
characterization of Darboux transformations obtained by Deift [3] (see [15]
for a further extension to the double-commutator method) to provide a novel
demonstration of the completeness of the exceptional polynomials. Second, we
characterize the possible gap sets of families of exceptional Hermite
polynomial in terms the corresponding partition.

##  2\. Some spectral theory

Let $\mathcal{R}\subset\mathbb{R}[x]$ be the set of real-valued polynomials
that have no real zeros. Consider a SLP on $(-\infty,\infty)$ of the form

(4) $-(W_{\eta}y^{\prime})^{\prime}-R_{\eta}y=\varepsilon
W_{\eta}y,\qquad\eta\in\mathcal{R},$

where

(5) $\displaystyle W_{\eta}$ $\displaystyle:=\eta^{-2}e^{-x^{2}},$ (6)
$\displaystyle R_{\eta}$
$\displaystyle:=\eta^{-3}(\eta^{\prime\prime}+2x\eta^{\prime})e^{-x^{2}}$

In this section we will consider Darboux transformations of self-adjoint
operators corresponding to SLP belonging to the class shown in (4).

Fix $\xi,\eta\in\mathcal{R}$ and set

(7) $\displaystyle\tilde{\eta}$ $\displaystyle:=\eta e^{x^{2}},$ (8)
$\displaystyle\hat{\eta}$ $\displaystyle:=\eta^{-1}e^{-\frac{x^{2}}{2}},$

with $\tilde{\xi},\hat{\xi}$ defined analogously.

Next, introduce the bilinear differential expression

(9)
$\operatorname*{\operatorname{\chi}}\\!\left(f,g\right)=fg^{\prime\prime}-2f^{\prime}g^{\prime}+f^{\prime\prime}g-2x(fg^{\prime}-f^{\prime}g).$

and define the following first, and second order differential expressions

(10) $\displaystyle\alpha_{\xi,\eta}y$
$\displaystyle=\eta^{-1}\operatorname{Wr}(\xi,y)$ (11)
$\displaystyle\beta_{\eta,\xi}$
$\displaystyle=\alpha_{\tilde{\eta},\tilde{\xi}}$ (12)
$\displaystyle\tau_{\eta}y$
$\displaystyle=\eta^{-1}\operatorname*{\operatorname{\chi}}\\!\left(\eta,y\right).$

with $\tau_{\xi}$ defined analogously. Note that $\tau_{\eta}$ is precisely
the second-order differential expression in the left-side of (3).

A direct calculation shows that

(13)
$(\alpha_{\xi,\eta}f)gW_{\xi}+f(\beta_{\eta,\xi}g)W_{\eta}=(\hat{\eta}f\,\hat{\xi}g)^{\prime},$

with $W_{\eta},W_{\xi}$ as defined in (5). Consequently, $\alpha_{\xi,\eta}$
and $-\beta_{\eta,\xi}$ are formally adjoint with respect to weights
$W_{\xi},W_{\eta}$. To be precise, if $I\subset\mathbb{R}$ is a compact
interval, and $f,g$ sufficiently smooth functions defined on $I$, we have

(14)
$\int_{I}(\alpha_{\xi,\eta}f)\bar{g}\,W_{\xi}+\int_{I}f(\beta_{\eta,\xi}\bar{g}),W_{\eta}=\hat{\eta}f\,\hat{\xi}\bar{g}\Big{|}_{I}$

Let us also recall Lagrange’s identity

$(\tau_{\eta}f)g\,W_{\eta}-f(\tau_{\eta}g)\,W_{\eta}=W_{\eta}\operatorname{Wr}\\!\left(f,g\right).$

Consequently, $\tau_{\eta}$ is symmetric with respect to $W_{\eta}$ in the
sense that

(15)
$\int_{I}(\tau_{\eta}f)\bar{g}W_{\eta}-\int_{I}f(\tau_{\eta}\bar{g})\,W_{\eta}=W_{\eta}\operatorname{Wr}\\!\left(f,\bar{g}\right)\Big{|}_{I},$

An analogous relation holds for $\tau_{\xi}$ and $W_{\xi}$.

Let
$(\mathcal{H}_{\eta},\left\langle\cdot,\cdot\right\rangle_{\eta})=\operatorname{\mathrm{L}}^{2}(\mathbb{R},W_{\eta})$
be the Hilbert space of square-integrable complex-valued functions defined on
$(-\infty,\infty)$, with $\mathcal{H}_{\xi}$ defined analogously. Let
${\mathcal{D}_{0}}$ denote the vector space of smooth, functions with compact
support, and let

(16)
$\mathcal{D}_{\xi}:=\\{f\in\mathcal{C}^{\infty}(\mathbb{R})\colon\operatorname{Wr}\\!\left(f,\xi\right)(x)\equiv
0\text{ for $|x|$ sufficiently large}\\}.$

In other words, an element of $\mathcal{D}_{\xi}$ behaves like a multiple of
$\xi(x)$ outside a compact interval, but with the constant of proportionality
for large $x$ not necessarily equal to the constant of proportionality for
small $x$. It is well known that $\mathcal{D}_{0}$ is dense in
$\mathcal{H}_{\eta}$ and $\mathcal{H}_{\xi}$. Since elements of
$\mathcal{D}_{\xi}$ are square integrable with respect to $W_{\eta}$ and since
$\mathcal{D}_{0}\subset\mathcal{D}_{\xi}$, the latter is also a dense
subspace.

Let
$a_{\xi,\eta}\colon\mathcal{H}_{\eta}\to\mathcal{H}_{\xi},\;b_{\eta,\xi}\colon\mathcal{H}_{\xi}\to\mathcal{H}_{\eta}$
denote densely defined first-order operators with action
$\alpha_{\xi,\eta},\beta_{\eta,\xi}$, respectively, and with

$\operatorname*{\operatorname{Dom}}a_{\xi,\eta}=\mathcal{D}_{\xi},\quad\operatorname*{\operatorname{Dom}}b_{\eta,\xi}=\mathcal{D}_{0}.$

Since $\alpha_{\xi,\eta}\xi\equiv 0$, it follows that
$\operatorname*{\operatorname{Ran}}a_{\xi,\eta}=\mathcal{D}_{0}$. For same
reason and by (14), we have
$\operatorname*{\operatorname{Ran}}b_{\eta,\xi}\subset\operatorname{Ann}_{\eta}\xi$,
where

(17) $\operatorname{Ann}_{\eta}\xi=\\{f\in\mathcal{H}_{\eta}\colon\left\langle
f,\xi\right\rangle_{\eta}=0.\\}.$

Next, we strengthen these assertions as follows.

######  Proposition 2.1.

We have
$\overline{\operatorname*{\operatorname{Ran}}a_{\xi,\eta}}=\mathcal{H}_{\xi}$.

###### Proof.

Since $\mathcal{D}_{0}$ is dense in $\mathcal{H}_{\xi}$, it suffices to show
that $\mathcal{D}_{0}\subset\operatorname*{\operatorname{Ran}}a_{\xi,\eta}$.
Let $h\in\mathcal{D}_{0}$ be given. Set

$f(x)=\xi(x)\int_{-\infty}^{x}h\eta\xi^{-2},\quad x\in\mathbb{R}.$

Since $h\in\mathcal{D}_{0}$, for $x$ sufficiently small, we have $f(x)=0$. For
$x$ sufficiently large, we have $f(x)=C\xi(x)$ where
$C=\int_{\mathbb{R}}h\eta\xi^{-2}.$ Hence, $f\in\mathcal{D}_{\xi}$. By (10) we
have,

$\alpha_{\xi,\eta}f=\eta^{-1}\xi^{2}\left(\xi^{-1}f\right)^{\prime}=h.$

∎

######  Proposition 2.2.

We have
$\overline{\operatorname*{\operatorname{Ran}}b_{\eta,\xi}}=\operatorname{Ann}_{\eta}\xi$.

###### Proof.

Let $\\{f_{n}\in\mathcal{D}_{0}\\}_{n\in\mathbb{N}}$ be a sequence such that
$\beta_{\eta,\xi}f_{n}\stackrel{{\scriptstyle n}}{{\longrightarrow}}h$ for
some $h\in\mathcal{H}_{\eta}$. Since $\alpha_{\xi,\eta}\xi=0$, by (13), we
have

$\left\langle\beta_{\eta,\xi}f_{n},\xi\right\rangle_{\eta}=0,\quad
n\in\mathbb{N}.$

Moreover, since

$\left\langle h,\xi\right\rangle_{\eta}=\left\langle
h-\beta_{\eta,\xi}f_{n},\xi\right\rangle_{\eta}\stackrel{{\scriptstyle
n}}{{\longrightarrow}}0,$

we have $h\in\operatorname{Ann}_{\eta}\xi$. Hence,
$\overline{\operatorname*{\operatorname{Ran}}b_{\eta,\xi}}\subseteq\operatorname{Ann}_{\eta}\xi$.

We now prove the converse. Let $h\in\operatorname{Ann}_{\eta}\xi$ be given.
Let $\\{h_{n}\in\mathcal{D}_{0}\\}_{n\in\mathbb{N}}$ be a sequence such that
$\|h_{n}-h\|\stackrel{{\scriptstyle n}}{{\longrightarrow}}0$. Choose a
$p\in\mathcal{D}_{0}$ such that $\left\langle p,\xi\right\rangle\neq 0$, and
set

$g_{n}=h_{n}-\frac{\left\langle h_{n},\xi\right\rangle_{\eta}}{\left\langle
p,\xi\right\rangle_{\eta}}p\in\mathcal{D}_{0},\quad n\in\mathbb{N}.$

By construction, $g_{n}\in\operatorname{Ann}_{\eta}\xi,\;n\in\mathbb{N}$. By
assumption, $\left\langle h_{n},\xi\right\rangle_{\eta}\stackrel{{\scriptstyle
n}}{{\longrightarrow}}0$. Hence,

$\|g_{n}-h\|_{\eta}\leq\|h_{n}-h\|_{\eta}+\left|\frac{\left\langle
h_{n},\xi\right\rangle_{\eta}}{\left\langle
p,\xi\right\rangle_{\eta}}\right|\|p\|_{\eta}\stackrel{{\scriptstyle
n}}{{\longrightarrow}}0.$

For $n\in\mathbb{N}$, set

$f_{n}(x)=\tilde{\eta}(x)\int_{-\infty}^{x}g_{n}\tilde{\xi}\tilde{\eta}^{-2}=\tilde{\eta}(x)\int_{-\infty}^{x}g_{n}\xi
W_{\eta},\quad x\in\mathbb{R}.$

Since $g_{n}$ has compact support, $f_{n}(x)=0$ for $x$ sufficiently small.
Since $g_{n}\in\operatorname{Ann}_{\eta}\xi$, we also have $f_{n}(x)=0$ for
$x$ sufficiently large. Hence, $f_{n}\in\mathcal{D}_{0}$. Observe that

$\beta_{\eta,\xi}f_{n}=\tilde{\xi}^{-1}\tilde{\eta}^{2}(\tilde{\eta}^{-1}f_{n})^{\prime}=g_{n}.$

Hence
$g_{n}\in\operatorname*{\operatorname{Ran}}b_{\eta,\xi},\;n\in\mathbb{N}$. We
already showed that $g_{n}\stackrel{{\scriptstyle n}}{{\longrightarrow}}h$.
This proves that

$\operatorname{Ann}_{\eta}\xi\subseteq\overline{\operatorname*{\operatorname{Ran}}b_{\eta,\xi}}.$

∎

Let $A_{\xi,\eta},B_{\eta,\xi}$ denote the maximal extensions of
$a_{\xi,\eta},b_{\eta,\xi}$, respectively. Formally,

(18) $\displaystyle\operatorname*{\operatorname{Dom}}A_{\xi,\eta}$
$\displaystyle=\\{f\in\mathcal{H}_{\eta}\colon
f\in\operatorname{AC}_{\mathrm{loc}}(\mathbb{R});\alpha_{\xi,\eta}f\in\mathcal{H}_{\xi}\\},$
$\displaystyle\operatorname*{\operatorname{Dom}}B_{\eta,\xi}$
$\displaystyle=\\{g\in\mathcal{H}_{\xi}\colon
g\in\operatorname{AC}_{\mathrm{loc}}(\mathbb{R});\beta_{\eta,\xi}g\in\mathcal{H}_{\eta}\\}.$

######  Proposition 2.3.

For $f\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}$ and
$g\in\operatorname*{\operatorname{Dom}}B_{\eta,\xi}$, we have

(19) $\left\langle A_{\xi,\eta}f,g\right\rangle_{\xi}=\left\langle
f,B_{\eta,\xi}g\right\rangle_{\eta}.$

###### Proof.

By (14), we have

$\hat{\eta}f\hat{\xi}g\Big{|}^{t}_{-t}\to\left\langle\alpha_{\xi,\eta}f,g\right\rangle_{\eta}-\left\langle
f,\beta_{\eta,\xi}g\right\rangle_{\xi}\quad\text{as }t\to\infty,$

with $\hat{\eta},\hat{\xi}$ as per (8). Observe that

$(\hat{\eta}f)^{2}=f^{2}W_{\eta},\quad(\hat{\xi}g)^{2}=g^{2}W_{\xi}.$

Hence, $\hat{\eta}f,\;\hat{\xi}g\in\operatorname{\mathrm{L}}^{2}(\mathbb{R})$.
Therefore, the above limit must be zero. ∎

######  Proposition 2.4.

We have $b_{\eta,\xi}^{*}\subseteq-A_{\xi,\eta}$ and
$a_{\xi,\eta}^{*}\subseteq-B_{\eta,\xi}$.

###### Proof.

Let
$f\in\operatorname*{\operatorname{Dom}}b_{\eta,\xi}^{*}\subset\mathcal{H}_{\eta}$.
By definition, there exists an $h\in\mathcal{H}_{\xi}$ such that

$\left\langle f,\beta_{\eta,\xi}g\right\rangle_{\eta}=\left\langle
h,g\right\rangle_{\xi},\quad\text{for all }g\in\mathcal{D}_{0}.$

Set

$\tilde{f}(x):=\xi\int_{0}^{x}h\eta\xi^{-2},\quad x\in\mathbb{R}.$

Hence, by (10), we have $\alpha_{\xi,\eta}[\tilde{f}]=h$ almost everywhere. By
(14), it then follows that

$\int_{\mathbb{R}}\tilde{f}\,(\beta_{\eta,\xi}\bar{g})\,W_{\eta}=\left\langle
h,g\right\rangle_{\xi},\quad\text{for all }g\in\mathcal{D}_{0}.$

Hence,

$\int_{\mathbb{R}}(f-\tilde{f})(\beta_{\eta,\xi}\bar{g})W_{\eta}=0\text{ for
all }g\in\mathcal{D}_{0}.$

Now, $f-\tilde{f}\in\operatorname{\mathrm{L}}^{2}(I,W_{\eta})$ for every
compact interval $I\subset\mathbb{R}$. Hence, by Proposition 2.2, $f$ and
$\tilde{f}$ differ by a multiple of $\xi$ a.e. on $I$. Since this is true for
every $I$, it follows that $f-\tilde{f}$ is a multiple of $\xi$ a.e. on all of
$\mathbb{R}$. Therefore, $f\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}$
with $\alpha_{\xi,\eta}f=h$.

Next, let $f\in\operatorname*{\operatorname{Dom}}a_{\xi,\eta}^{*}$. By
definition, there exists an $h\in\mathcal{H}_{\eta}$ such that

$\left\langle f,\alpha_{\xi,\eta}g\right\rangle_{\xi}=\left\langle
h,g\right\rangle_{\eta},\quad\text{for all
}g\in\mathcal{D}_{\xi}\subset\mathcal{H}_{\eta}.$

If $g=\xi$, then the above expressions vanish. Hence,
$h\in\operatorname{Ann}_{\eta}\xi$, which implies that

$\tilde{f}(x):=\tilde{\eta}(x)\int_{-\infty}^{x}h\xi W_{\eta},\quad
x\in\mathbb{R}$

is well defined. By construction, $\beta_{\eta,\xi}\tilde{f}=h$ a.e. It
follows that

$\int_{\mathbb{R}}\tilde{f}(\alpha_{\xi,\eta}\bar{g})\,W_{\xi}=\left\langle
h,g\right\rangle_{\eta},\quad\text{for all }g\in\mathcal{D}_{\xi}.$

Hence,

$\int_{\mathbb{R}}(f-\tilde{f})(\alpha_{\xi,\eta}\bar{g})\,W_{\xi}=0\text{ for
all }g\in\mathcal{D}_{\xi}.$

Therefore, by Proposition 2.1, $f=\tilde{f}$ a.e. on $\mathbb{R}$. ∎

######  Proposition 2.5.

We have $A_{\xi,\eta}=-B_{\eta,\xi}^{*}$ and $-B_{\eta,\xi}=A_{\xi,\eta}^{*}$.

###### Proof.

By Proposition 2.3, we have $A_{\xi,\eta}\subseteq-B_{\eta,\xi}^{*}$ and
$-B_{\eta,\xi}\subseteq A_{\xi,\eta}^{*}$. By Proposition 2.4,

$-B_{\eta,\xi}^{*}\subseteq-b_{\eta,\xi}^{*}\subseteq
A_{\xi,\eta},\quad\text{and}\quad A_{\xi,\eta}^{*}\subseteq
a_{\xi,\eta}^{*}\subseteq-B_{\eta,\xi},$

as was to be shown. ∎

Let $T_{\eta}:\mathcal{H}_{\eta}\to\mathcal{H}_{\eta}$ denote the densely
defined operator with action $\tau_{\eta}$ and with maximal domain

(20)
$\operatorname*{\operatorname{Dom}}(T_{\eta})=\\{f\in\operatorname{AC}_{\mathrm{loc}}^{1}(\mathbb{R})\colon\tau_{\eta}f\in\mathcal{H}_{\eta}\\}$

Let $T_{\xi}:\mathcal{H}_{\xi}\to\mathcal{H}_{\xi}$ be the analogous operator
with action $\tau_{\xi}$.

######  Proposition 2.6.

Let $\eta,\xi\in\mathcal{R}$ be such that

(21)
$\operatorname*{\operatorname{\chi}}\\!\left(\eta,\xi\right)=\varepsilon_{0}\eta\xi,\quad\varepsilon_{0}\in\mathbb{R}.$

Then,

(22) $\displaystyle T_{\eta}$
$\displaystyle=B_{\eta,\xi}A_{\xi,\eta}+\varepsilon_{0},$ $\displaystyle
T_{\xi}$ $\displaystyle=A_{\xi,\eta}B_{\eta,\xi}+\varepsilon_{0}+2$

in the the sense of naturally defined composition; i.e.

$\displaystyle\operatorname*{\operatorname{Dom}}B_{\eta,\xi}A_{\xi,\eta}$
$\displaystyle=\\{f\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}\colon
A_{\xi,\eta}f\in\operatorname*{\operatorname{Dom}}B_{\eta,\xi}\\};$
$\displaystyle\operatorname*{\operatorname{Dom}}A_{\xi,\eta}B_{\eta,\xi}$
$\displaystyle=\\{f\in\operatorname*{\operatorname{Dom}}B_{\eta,\xi}\colon
B_{\eta,\xi}f\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}\\}.$

######  Lemma 2.1.

Relation (21) is equivalent to either of the following:

(23) $\displaystyle\tau_{\eta}\xi$ $\displaystyle=\varepsilon_{0}\xi,$ (24)
$\displaystyle\tau_{\xi}\tilde{\eta}$
$\displaystyle=(\varepsilon_{0}+2)\tilde{\eta}.$

###### Proof.

The equivalence of (21) and (23) follows directly from the definition (12).
The following differential relation can be verified by direct calculation:

(25)
$\operatorname*{\operatorname{\chi}}\\!\left(\xi,\tilde{\eta}\right)=e^{x^{2}}\operatorname*{\operatorname{\chi}}\\!\left(\eta,\xi\right)+2\xi\tilde{\eta}$

Hence,

$\tilde{\eta}\tau_{\eta}\xi=e^{x^{2}}\operatorname*{\operatorname{\chi}}\\!\left(\eta,\xi\right)=\operatorname*{\operatorname{\chi}}\\!\left(\xi,\tilde{\eta}\right)-2\xi\tilde{\eta}=\xi(\tau_{\xi}-2)\tilde{\eta}.$

This proves the equivalence of (23) and (24). ∎

######  Lemma 2.2.

Suppose that (21) holds. Then,

(26) $\displaystyle\tau_{\eta}$
$\displaystyle=\beta_{\eta,\xi}\alpha_{\xi,\eta}+\varepsilon_{0},$ (27)
$\displaystyle\tau_{\xi}$
$\displaystyle=\alpha_{\xi,\eta}\beta_{\eta,\xi}+\varepsilon_{0}+2$

###### Proof.

By definition and direct calculation,

(28)
$\operatorname*{\operatorname{\chi}}\\!\left(\eta,f\right)\xi-\operatorname*{\operatorname{\chi}}\\!\left(\eta,\xi\right)f=\tilde{\eta}^{-1}\operatorname{Wr}\\!\left(\eta\tilde{\eta},\operatorname{Wr}\\!\left(\xi,f\right)\right)$

Hence, by (10) - (12),

$\displaystyle\tau_{\eta}f$
$\displaystyle=(\tilde{\xi}\eta^{2})^{-1}\operatorname{Wr}\\!\left(\eta\tilde{\eta},\operatorname{Wr}\\!\left(\xi,f\right)\right)+\xi^{-1}f\tau_{\eta}\xi$
$\displaystyle=\tilde{\xi}^{-1}\operatorname{Wr}\\!\left(\tilde{\eta},\eta^{-1}\operatorname{Wr}\\!\left(\xi,f\right)\right)+\varepsilon_{0}f$
$\displaystyle=\beta_{\eta,\xi}\alpha_{\xi,\eta}f+\varepsilon_{0}f$

By definition and direct calculation,

$\displaystyle\alpha_{\tilde{\eta},\xi}f$
$\displaystyle=e^{x^{2}}\beta_{\eta,\xi}f,$
$\displaystyle\beta_{\xi,\tilde{\eta}}f$
$\displaystyle=e^{-2x^{2}}\eta^{-1}\operatorname{Wr}\\!\left(e^{x^{2}}\xi,f\right)=\eta^{-1}\operatorname{Wr}\\!\left(\xi,e^{-x^{2}}f\right)=\alpha_{\xi,\eta}(e^{-x^{2}}f)$

Applying (26) with
$\eta\mapsto\xi,\;\xi\mapsto\tilde{\eta},\;\varepsilon_{0}\mapsto\varepsilon_{0}+2$,
and using (24) gives

$\displaystyle\tau_{\xi}$
$\displaystyle=\beta_{\xi,\tilde{\eta}}\alpha_{\tilde{\eta},\xi}+\varepsilon_{0}+2=\alpha_{\xi,\eta}\beta_{\eta,\xi}+\varepsilon_{0}+2.$

∎

###### Proof.

By Lemma 2.2, the formal factorization relations (26) and (27) hold. The
differential expressions $\tau_{\eta},\tau_{\xi}$ are limit point at
$\pm\infty$; see [2, Theorem 2.4]. Hence, $T_{\eta},T_{\xi}$ are self-adjoint.
By Proposition 2.5, $A_{\xi,\eta},-B_{\eta,\xi}$ are adjoint. Hence, the
natural compositions $A_{\xi,\eta}B_{\eta,\xi}$ and $B_{\eta,\xi}A_{\xi,\eta}$
are self-adjoint by a standard theorem in functional analysis; see for example
[27, Theorem 13.13]. Since there are no boundary conditions at $\pm\infty$ the
LHS and the RHS of (22) must coincide. ∎

######  Proposition 2.7.

Let $\eta,\xi\in\mathcal{R}$ such that (21) holds. Let
$\psi\in\mathcal{H}_{\eta}$ be an eigenfunction of $T_{\eta}$; i.e.
$\tau_{\eta}\psi=\varepsilon\psi,\;\varepsilon\in\mathbb{R}$. Then,
necessarily $\varepsilon_{0}>\varepsilon$ and

$\hat{\psi}:=\alpha_{\xi,\eta}\psi$

is an eigenfunction of $T_{\xi}$ with eigenvalue $\varepsilon+2$. Moreover,

(29)
$\|\hat{\psi}\|^{2}_{\xi}=(\varepsilon_{0}-\varepsilon)\|\psi\|_{\eta}^{2}$

###### Proof.

By assumption $\psi\in\operatorname*{\operatorname{Dom}}T_{\eta}$. Hence, by
Proposition 2.6, $\psi\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}$, and
$\hat{\psi}\in H_{\xi}$. By (19) and Proposition 2.6, we have

$\displaystyle\left\langle\hat{\psi},\hat{\psi}\right\rangle_{\xi}$
$\displaystyle=-\left\langle
B_{\eta,\xi}A_{\xi,\eta}\psi,\psi\right\rangle_{\eta}$
$\displaystyle=\left\langle(\varepsilon_{0}-T_{\eta})\psi,\psi\right\rangle_{\eta}=(\varepsilon_{0}-\varepsilon)\left\langle\psi,\psi\right\rangle_{\eta}.$

By construction,

$\displaystyle\beta_{\eta,\xi}\hat{\psi}$
$\displaystyle=\beta_{\eta,\xi}\alpha_{\xi,\eta}\psi=(\varepsilon-\varepsilon_{0})\psi\in\mathcal{H}_{\eta},$
$\displaystyle\alpha_{\xi,\eta}\beta_{\eta,\xi}\hat{\psi}$
$\displaystyle=(\varepsilon-\varepsilon_{0})\hat{\psi}\in\mathcal{H}_{\xi}.$

Hence, $\hat{\psi}\in\operatorname*{\operatorname{Dom}}B_{\eta,\xi}$ and
$\beta_{\eta,\xi}\hat{\psi}\in\operatorname*{\operatorname{Dom}}A_{\xi,\eta}$.
It follows that, $\hat{\psi}\in\operatorname*{\operatorname{Dom}}T_{\xi}$,
with

$T_{\xi}\hat{\psi}=(A_{\xi,\eta}B_{\eta,\xi}+\varepsilon_{0}+2)\hat{\psi}=(\varepsilon+2)\hat{\psi}.$

∎

######  Theorem 2.1.

Let $\eta,\xi\in\mathcal{R}$ be such that (21) holds. Let
$\sigma(T_{\eta}),\sigma(T_{\xi})$ denote the spectral sets of the indicated
operators. Then, $\varepsilon_{0}=\max\sigma(T_{\eta})$. Moreover,

$\sigma(T_{\xi}-2)=\sigma(T_{\eta})\setminus\\{\varepsilon_{0}\\}.$

###### Proof.

Both $T_{\eta}$ and $T_{\xi}$ have a pure point spectrum; for a proof see
Section 5.9 of [29] or Problem 1 in Ch. 9 of [2]. By assumption,
$\psi_{0}:=\xi$ has no real zeros. Relation (21) is equivalent to

$\tau_{\eta}\psi_{0}=\varepsilon_{0}\psi_{0}.$

Hence, $\varepsilon_{0}=\max\sigma(T_{\eta})$ by the Sturm oscillation
theorem. Let $\psi_{n}\in\mathcal{H}_{\eta},\;n\in\mathbb{N}$ be the other
eigenfunctions of $T_{\eta}$ with

$\tau_{\eta}\psi_{n}=\varepsilon_{n}\psi_{n},\quad n=1,2,\ldots$

and $\varepsilon_{0}>\varepsilon_{1}>\varepsilon_{2}>\cdots$. Set

$\hat{\psi}_{n}=\alpha_{\xi,\eta}\psi_{n},\quad n=1,2,\ldots.$

By Proposition 2.7, these are all eigenfunctions of $T_{\xi}$ with eigenvalues
$\varepsilon_{n},\;n=1,2,\ldots$. By Proposition 2.1,
$\operatorname*{\operatorname{Ran}}A_{\xi,\eta}=\mathcal{H}_{\xi}$. Since
$\\{\psi_{n}\\}_{n\in{\mathbb{N}_{0}}}$ is complete in $\mathcal{H}_{\eta}$,
it follows that $\\{\hat{\psi}_{n}\\}_{n\in\mathbb{N}}$ is complete in
$\mathcal{H}_{\xi}$. Therefore
$\sigma(T_{\xi})=\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$. ∎

Deift [3] proved a more general version of Theorem 2.1 regarding the spectra
of general self-adjoint operators related by a factorization/Darboux
transformation. Nonetheless, for the sake of comprehensiveness we prefer to
state and prove a more restrictive version limited to Sturm-Liouville
operators. The above proof of Theorem 2.1 is more accessible than Deift’s more
abstract argument.

##  3\. The formal theory of Exceptional Hermite polynomials

Classical Hermite polynomials are orthogonal polynomials defined by the
recurrence relation

(30) $H_{0}=1,\quad xH_{n}=\frac{1}{2}H_{n+1}+nH_{n-1},\quad n=1,2,\ldots$

They satisfy the following orthogonality relation:

(31)
$\int_{-\infty}^{\infty}H_{m}(x)H_{n}(x)e^{-x^{2}}dx=\sqrt{\pi}\,2^{n}n!\delta_{n,m}$

Hermite polynomials are known as classical orthogonal polynomials, because
they also arise as solutions of the Hermite differential equation

(32) $y^{\prime\prime}-2xy^{\prime}=\varepsilon y$

with $y=H_{n},\varepsilon=-2n,\;n\in{\mathbb{N}_{0}}$. Note that
multiplication of (32) by $-e^{-x^{2}}$ gives a singular Sturm-Liouville
problem on $(-\infty,\infty)$, namely

$-(e^{-x^{2}}y)^{\prime}=-\varepsilon e^{-x^{2}}y.$

This SLP is limit-point at $\pm\infty$, so no explicit boundary conditions are
required.

Exceptional Hermite polynomials are a generalization of the classical Hermite
polynomials because they satisfy a second-order, Hermite-like differential
equation. Each family of such polynomials is indexed by a partition, and so we
begin by recalling some relevant definitions.

######  Definition 3.1.

A partition $\lambda$ of a natural number $N$ is a non-increasing, finitely
supported sequence of non-negative integers
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq 0$ that sum to $N$. The length
$\ell$ of a partition $\lambda$ is defined to be the smallest
$\ell\in{\mathbb{N}_{0}}$ such that $\lambda_{\ell+1}=0$. Thus,
$N=\lambda_{1}+\cdots+\lambda_{\ell}$, with $\lambda_{\ell}>0$.

Fix a partition $\lambda$, and let

(33) $\displaystyle C_{\lambda,l}$ $\displaystyle:=2^{l(l-1)/2}\prod_{1\leq
i<j\leq l}(\lambda_{i}-\lambda_{j}+j-i)$ (34)
$\displaystyle\pi_{\lambda,l}(n)$
$\displaystyle:=\prod_{i=1}^{l}(n-N-\lambda_{i}+i),\quad k\in\mathbb{N},$ (35)
$\displaystyle m_{i}$ $\displaystyle:=\lambda_{i}+\ell-i,\quad
i\in\mathbb{N};$ (36) $\displaystyle\eta_{\lambda}$
$\displaystyle:=(C^{\lambda}_{\ell})^{-1}\operatorname{Wr}\\!\left(H_{m_{\ell}},\dots,H_{m_{1}}\right),$
(37) $\displaystyle{I_{\lambda}}$
$\displaystyle:=\\{n\in{\mathbb{N}_{0}}\colon n-N+\ell\geq 0\text{ and
}\pi_{\lambda,\ell}(n)\neq 0.\\},$ (38) $\displaystyle H_{\lambda,n}$
$\displaystyle:=(2^{\ell}C^{\lambda}_{\ell}\pi_{\lambda,\ell}(n))^{-1}\operatorname{Wr}\\!\left(H_{m_{\ell}},\dots,H_{m_{1}},H_{n-N+\ell}\right),\quad
n\in{I_{\lambda}}$

where $\operatorname{Wr}$ denotes the usual Wronskian determinant.

Observe that $n-N+\ell\in\\{m_{1},\ldots,m_{\ell}\\}$ if and only if
$\pi_{\lambda,\ell}(n)=0$. Thus, $H_{\lambda,n},\;n\in{I_{\lambda}}$ is a non-
zero Wronskian of classical Hermite polynomials. Also note that if $N=0$ and
$\lambda$ is the empty partition, then $\eta_{\lambda}=1$ and
$H_{\lambda,n}=H_{n}$, is the classical $n$th degree Hermite polynomial.

######  Proposition 3.1.

Let $\eta_{\lambda},H_{\lambda,n},\;n\in{I_{\lambda}}$ be as above. Then

(39) $\displaystyle\eta_{\lambda}(x)$ $\displaystyle=2^{N}x^{N}+\text{ lower
degree terms.}$ (40) $\displaystyle H_{\lambda,n}(x)$
$\displaystyle=2^{n}x^{n}+\text{ lower degree terms.}$

###### Proof.

Let $p_{i},\;i=1,\ldots,l$ be a polynomial of degree $d_{i}$ with leading
coefficient $c_{i}$. Suppose that $d_{1},\ldots,d_{l}$ are distinct and let
$P=\operatorname{Wr}\\!\left(p_{1},\ldots,p_{l}\right)$. Then

$\deg P=\sum_{i=1}^{l}d_{i}-\frac{1}{2}l(l-1)=\sum_{i=1}^{l}(d_{i}-l+i).$

Consequently,

$\displaystyle\deg\eta_{\lambda}$
$\displaystyle=\sum_{i=1}^{\ell}(m_{i}-\ell+i)=\sum_{i}\lambda_{i}=N,$
$\displaystyle\deg H_{\lambda,n}$
$\displaystyle=n-N+\ell-(\ell+1)+\ell+1+\sum_{i=1}^{\ell}(m_{i}-\ell-1+i)=n.$

The leading coefficient of $P$ is given by $\prod_{i=1}^{k}c_{i}\prod_{1\leq
i<j\leq k}(d_{j}-d_{i})$. It follows that $C_{\lambda,\ell}$ and
$2^{\ell}C_{\lambda,\ell}\pi_{\lambda,\ell}(n)$ are precisely the leading
coefficients of $\eta_{\lambda}$ and $H_{\lambda,n}$, respectively. ∎

The formulation of $\eta_{\lambda}$ and $H_{\lambda,n}$ in (36) and (38) is
based on Wronskians of size $\ell$ and $\ell+1$ respectively. However, it is
possible to define these polynomials using Wronskians of any size $l\geq\ell$.

######  Proposition 3.2.

Let $\lambda$ be a partition of $N$ with length $\ell$. Fix $l\geq\ell$, and
set

$m_{i,l}=\lambda_{i}+l-i,\quad i=1,\ldots,l.$

Then,

(41) $\displaystyle\eta_{\lambda}$
$\displaystyle=C_{\lambda,l}^{-1}\operatorname{Wr}\\!\left(H_{m_{l,l}},\ldots,H_{m_{1,l}}\right),$
(42) $\displaystyle H_{\lambda,n}$
$\displaystyle=(2^{l}C^{\lambda}_{l}\pi_{\lambda,l}(n))^{-1}\operatorname{Wr}\\!\left(H_{m_{l,l}},\dots,H_{m_{l,1}},H_{n-N+l}\right)$

###### Proof.

The proof is by induction on $l-\ell$. If $l=\ell$, there is nothing to prove.
Suppose that (41) holds for a particular $l\geq\ell$. Observe that

$m_{i,l+1}=m_{i,l}+1,\quad i=1,\ldots,l,$

with $m_{l+1,l+1}=0$. Since

$H_{n}^{\prime}=2nH_{n-1},\quad n=1,2,\ldots,$

we have

$\displaystyle\operatorname{Wr}(H_{m_{l+1,l+1}},H_{m_{l+1,l}},\ldots,H_{m_{l+1,1}})$
$\displaystyle=\operatorname{Wr}(1,H_{m_{l,l}+1},\ldots,H_{m_{l,1}+1})$
$\displaystyle=\operatorname{Wr}(H_{m_{l,l}+1}^{\prime},\ldots,H_{m_{l,1}+1}^{\prime})$
$\displaystyle=2^{l}\prod_{i}(m_{l,i}+1)\operatorname{Wr}(H_{m_{l,l}},\ldots,H_{m_{l,1}}).$

Observe that

$\sum_{i=1}^{l+1}m_{l+1,i}=N+l(l+1)/2,$

the leading term coefficient of the left-side Wronskian is
$2^{N}C_{\lambda,l+1}$ , it follows that

$\operatorname{Wr}(H_{m_{l+1,l+1}},H_{m_{l+1,l}},\ldots,H_{m_{l+1,1}})=C_{\lambda,l+1}\eta_{\lambda}.$

Relation (42) is proved in an anolgous fashion. ∎

######  Corollary 3.1.

For every $l\geq\ell$ and $n\in{I_{\lambda}}$, we have

(43)
$\frac{H_{\lambda,n}}{\eta_{\lambda}}=\frac{\operatorname{Wr}\\!\left(H_{m_{l,l}},\ldots,H_{m_{1,l}},H_{n-N+l}\right)}{2^{l}\pi_{\lambda,l}(n)\operatorname{Wr}\\!\left(H_{m_{l,l}},\ldots,H_{m_{1,l}}\right)}.$

In other words, the rational function $\eta_{\lambda}^{-1}H_{\lambda,n}$ can
be defined without invoking the normalizing constant $C_{\lambda,l}$ of (33).

A particularly interesting case of this formulation occurs when we take $l=N$.
By (40), the set ${I_{\lambda}}$ is simply the degree sequence of the
polynomial family $\\{H_{\lambda,n}\\}_{n}$. Let

(44) $\displaystyle k_{i}$ $\displaystyle:=m_{i,N}=\lambda_{i}+N-i,\quad
i=1,\ldots,N,$ (45) $\displaystyle{K_{\lambda}}$
$\displaystyle:=\\{k_{1},\ldots,k_{N}\\};$

and observe that

(46) ${K_{\lambda}}=\\{0,1,\ldots,N-\ell-1\\}\cup\\{m_{i}+N-\ell\colon
i=1,\ldots,\ell\\}.$

Recall that $n-N+\ell\in\\{m_{1},\ldots,m_{\ell}\\}$ if and only if
$\pi_{\lambda,\ell}(n)=0$. It follows that
${K_{\lambda}}={\mathbb{N}_{0}}\setminus{I_{\lambda}}$ is the set of
exceptional degrees missing from ${I_{\lambda}}$, and that there are precisely
$N=\deg\eta_{\lambda}$ such “exceptional” degrees. Moreover, by (41) and (42),
we have

(47) $\displaystyle\eta_{\lambda}$
$\displaystyle\propto\operatorname{Wr}\\!\left(H_{k_{N}},\ldots,H_{k_{1}}\right),$
$\displaystyle H_{\lambda,n}$
$\displaystyle\propto\operatorname{Wr}\\!\left(H_{k_{N}},\ldots,H_{k_{1}},n\right),\quad
n\in{I_{\lambda}}.$

Thus both $\eta_{\lambda}$ and the exceptional polynomials $H_{\lambda,n}$ can
be formulated as Wronskians involving the exceptional degrees. Of course, for
most partitions $N>\ell$ and so the formulations in (36) and (38) are more
economical. See [9] for a characterization of the smallest determinant that
can be used to represent a Wronskian of Hermite polynomials.

Just like their classical counterparts, exceptional Hermite polynomials
satisfy a second-order eigenvalue relation. Set

(48)
${\tau_{\lambda}}y:=y^{\prime\prime}-2\left(x+\frac{\eta_{\lambda}^{\prime}}{\eta_{\lambda}}\right)y^{\prime}+\left(\frac{\eta_{\lambda}^{\prime\prime}}{\eta_{\lambda}}+2x\frac{\eta_{\lambda}^{\prime}}{\eta_{\lambda}}\right)y;$

i.e., ${\tau_{\lambda}}$ is the second-order differential expression in the
left-side of (3) with $\eta=\eta_{\lambda}$.

######  Proposition 3.3.

Let $m_{1},\ldots,m_{l},m\in{\mathbb{N}_{0}}$ be distinct non-zero integers.
Set

$\displaystyle\eta$
$\displaystyle=\operatorname{Wr}\\!\left(H_{m_{l}},\ldots,H_{m_{1}}\right)$
$\displaystyle\xi$
$\displaystyle=\operatorname{Wr}\\!\left(H_{m_{l}},\ldots,H_{m_{1}},H_{m}\right)$

Then,

(49) $\chi(\eta,\xi)=2(l-m)\eta\xi.$

###### Proof.

Relation (49) is equivalent to

(50) $(\tau_{\eta}+2(m-l))\xi=0$

Set

$\tilde{\tau}_{\eta}y=y^{\prime\prime}-2xy^{\prime}+2(\log\eta)^{\prime\prime}y.$

An elementary calculation shows that

$\tau_{\eta}(\eta y)=\eta\tilde{\tau}_{\eta}y.$

Thus, it suffices to show that

(51) $(\tilde{\tau}_{\eta}+2(m-l))(\eta^{-1}\xi)=0.$

Let

$\tau_{\emptyset}y=y^{\prime\prime}-2xy^{\prime}$

be the classic Hermite differential expression. Define the $l$th order
differential expression

$\kappa
y=\frac{\operatorname{Wr}\\!\left(H_{m_{l}},\ldots,H_{m_{1}},y\right)}{\operatorname{Wr}\\!\left(H_{m_{l}},\ldots,H_{m_{1}}\right)}=y^{(l)}-(\log\eta)^{\prime}y^{(l-1)}+\cdots,$

and observe that

$\kappa H_{m}=\eta^{-1}\xi.$

Since

$\tau_{\emptyset}H_{m}=-2mH_{m},$

it suffices to show that

$(\tilde{\tau}_{\lambda}-2l)\kappa=\kappa\tau_{\emptyset},$

This, in turn, is equivalent to the relation

$[\tau_{\emptyset},\kappa]=(2l-2(\log\eta)^{\prime\prime})\kappa.$

By construction,
$\ker\kappa=\operatorname*{\operatorname{span}}\\{H_{m_{1}},\ldots,H_{m_{l}}\\}$.
Since these are all eigenfunctions of $\tau_{\emptyset}$, the commutator
$[\tau_{\emptyset},\kappa]$ annihilates $\ker\kappa$. By inspection,
$[\tau_{\emptyset},\kappa]$ has order $l$. Hence,
$[\tau_{\emptyset},\kappa]=f\kappa$ for some function $f(x)$. Observe that

$[\tau_{\emptyset},\kappa]y=(2l-2(\log\eta)^{\prime\prime})y^{(l)}+\cdots$

Hence, $f=2l-2(\log\eta)^{\prime\prime}$. This proves (51), which is
equivalent to (49). ∎

######  Corollary 3.2.

Let $\lambda$ be a partition and $n\in{I_{\lambda}}$ an allowed degree. Then,

(52)
$\operatorname*{\operatorname{\chi}}\\!\left(\eta_{\lambda},H_{\lambda,n}\right)=2(N-n)\eta_{\lambda}H_{\lambda,n}.$

###### Proof.

Since $\chi$ is bilinear, this follows by (47) and the preceding Proposition.
∎

The following converse was proved in [8].

######  Proposition 3.4.

Let $\eta(x)$ be a non-zero polynomial. Let $\\{y_{n}\colon\deg y=n\\}_{n\in
I}$ be the family of polynomial solutions of

$\chi(\eta,y)=\varepsilon\eta y\quad\varepsilon\in\mathbb{C}.$

Suppose that ${\mathbb{N}_{0}}\setminus I$ is finite; i.e., the family of
polynomial solutions is missing at most finitely many degrees. Then, there
exists a partition $\lambda$ such that $I={I_{\lambda}}$, and such that, up to
a multiplicative constant, $\eta=\eta_{\lambda}$ and
$y_{n}=H_{\lambda,n},\;n\in{I_{\lambda}}$.

###  3.1. Multi-step factorization chains.

In this section, we show that every exceptional operator $T_{\lambda}$ is
connected to the classical

$\tau_{\emptyset}y=y^{\prime\prime}-2xy^{\prime}$

by a finite factorization chain of exceptional Hermite operators. We will
describe two such chains: one will connect an exceptional Hermite operator
${\tau_{\lambda}}$ to $\tau_{\emptyset}$ and the other will connect
$\tau_{\emptyset}$ to ${\tau_{\lambda}}$.

Let $\lambda$ be a partition of $N$. Let $k_{1},\ldots,k_{N}\in{K_{\lambda}}$
be the exceptional degrees as per (44), and let
$n_{1},\ldots,n_{l}\in{I_{\lambda}}$ be a list of allowed degrees. Set

$\eta_{i}=\operatorname{Wr}\\!\left(H_{k_{N}},\ldots,H_{k_{1}},H_{n_{1}},\ldots,H_{n_{i}}\right),\quad
i=0,1,\ldots,l,$

and introduce the differential expressions

$\displaystyle\alpha_{i}$ $\displaystyle=\alpha_{\eta_{i+1},\eta_{i}},\quad
i=0,\ldots,l-1,$ $\displaystyle\beta_{i}$
$\displaystyle=\alpha_{\eta_{i},\eta_{i+1}}$ $\displaystyle\tau_{i}$
$\displaystyle=\tau_{\eta_{i}},\quad i=0,\ldots,l.$

In particular, $\tau_{0}={\tau_{\lambda}}$.

######  Proposition 3.5.

With the above definitions, we have

(53)
$\displaystyle\operatorname*{\operatorname{\chi}}\\!\left(\eta_{i},\eta_{i+1}\right)$
$\displaystyle=2(N+i-n_{i+1})\eta_{i}\eta_{i+1},\quad i=0,\ldots,l-1,$ (54)
$\displaystyle\tau_{i}$
$\displaystyle=\beta_{i}\alpha_{i}+2(N+i-n_{i+1}),\quad i=0,\ldots,l-1;$ (55)
$\displaystyle\tau_{i+1}$
$\displaystyle=\alpha_{i}\beta_{i}+2(N+i+1-n_{i+1}).$

###### Proof.

Relation (53) follows by Corollary 3.2. Relations (54) and (55) follow by
Lemma 2.2. ∎

Next, we describe a factorization chain that connects $\tau_{\lambda}$ to
$\tau_{\emptyset}$ using $\lambda_{1}$ steps. Let
$k_{1},k_{2},\ldots,k_{N}\in{K_{\lambda}}$ be the exceptional degrees, as per
(44). Since $k_{1}=\lambda_{1}+N-1$, every $n\geq\lambda_{1}+N$ is an allowed
degree; i.e. an element of ${I_{\lambda}}$.

######  Definition 3.2.

We call the smallest $\lambda_{1}$ elements of ${I_{\lambda}}$ the _sporadic_
degrees of ${I_{\lambda}}$.

######  Proposition 3.6.

An $n\in{I_{\lambda}}$ is a sporadic degree if and only if there exists an
exceptional degree $k\in{K_{\lambda}}$ such that $n<k$.

###### Proof.

By (44), the $N$ exceptional degrees are contained in
$\\{0,1,\ldots,\lambda_{1}+N-1\\}$, a set of cardinality $\lambda_{1}+N$.
Hence, $n\in{I_{\lambda}}$ is dominated by some exceptional degree if and only
if $n<\lambda_{1}+N-1$. By the preceding remark, there are precisely
$\lambda_{1}$ such degrees. ∎

######  Proposition 3.7.

Let $\lambda$ be a partition of $N$, and let $n_{1}<n_{2}<\ldots$ be the
allowed degrees in ${I_{\lambda}}$ listed in increasing order. For
$j\in{\mathbb{N}_{0}}$, set

(56) $\displaystyle\lambda^{(j)}_{i}$
$\displaystyle=\max(\lambda_{i}-j,0)\quad i\in\mathbb{N},$

and let $\ell_{j}$ be the length of $\lambda^{(j)}$. Then,

(57) $\ell_{j}=N-n_{j+1}+j,\quad j\in{\mathbb{N}_{0}}.$

###### Proof.

Observe that if $j\geq\lambda_{1}$, then $\lambda^{(j)}=\emptyset$ is the
trivial partition. Thus it suffices to establish (57) for
$j=0,\ldots,\lambda_{1}-1$.

By definition, $i\leq\ell_{j}$ if and only if $\lambda_{i}-j\geq 1$. In other
words, the sequence $(\ell_{0},\ell_{1},\ell_{2},\ldots)$ and $\lambda$ are
dual partitions. Observe that if $\lambda_{i}-j\geq 1$, then $i-\ell_{j}\leq
0$. Conversely, $\lambda_{i}-j\leq 0$ if and only if $i-\ell_{j}\geq 1$.
Hence,

$\lambda_{i}-i\neq j-\ell_{j},\quad i\in\mathbb{N},\;j\in{\mathbb{N}_{0}}.$

Using $\sqcup$ for disjoint union, it follows that

$\\{\lambda_{i}-i\colon i=1,\ldots,N\\}\sqcup\\{j-\ell_{j}\colon
j=0,\ldots,\lambda_{1}-1\\}=\\{-N,-N+1,\ldots,\lambda_{1}-1\\}.$

Therefore,

$\\{j-\ell_{j}+N\colon j=0,\ldots,\lambda_{1}-1\\}$

is precisely the set of sporadic degrees in ${I_{\lambda}}$. Relation (57)
follows. ∎

######  Proposition 3.8.

Let $\lambda$ be a partition, with $\lambda^{(j)},\;j\in{\mathbb{N}_{0}}$ the
sequence of partitions defined by (56). Define the differential expressions

$\displaystyle\alpha_{j}$
$\displaystyle=\alpha_{\eta_{\lambda^{(j+1)}},\eta_{\lambda^{(j)}}},\quad
j=0,\ldots,\lambda_{1}-1,$ $\displaystyle\beta_{j}$
$\displaystyle=\alpha_{\eta_{\lambda^{(j)}},\eta_{\lambda^{(j+1)}}};$
$\displaystyle\tau_{j}$ $\displaystyle=\tau_{\eta_{\lambda^{(j)}}},\quad
j=0,\ldots,\lambda_{1}.$

Then,

(58) $\displaystyle\tau_{j}$
$\displaystyle=\beta_{j}\alpha_{j}+2\ell_{j},\quad j=0,\ldots,\lambda_{1}-1;$
(59) $\displaystyle\tau_{j+1}$
$\displaystyle=\alpha_{j}\beta_{j}+2(\ell_{j}+1).$

###### Proof.

Let $n_{1}<\ldots<n_{\lambda_{1}}$ be the sporadic degrees of $\lambda$ listed
in increasing order. Let

$\displaystyle K_{j}$
$\displaystyle={K_{\lambda}}\cup\\{n_{1},\ldots,n_{j}\\},\quad
j=0,\ldots,\lambda_{1},$ $\displaystyle\eta_{j}$
$\displaystyle=\operatorname{Wr}\\!\left(k_{N},\ldots,k_{1},n_{1},\ldots,n_{j}\right).$

By (57) and (53),

$\operatorname*{\operatorname{\chi}}\\!\left(\eta_{j},\eta_{j+1}\right)=2(N+j-n_{j+1})\eta_{j}\eta_{j+1}=2\ell_{j}\eta_{j}\eta_{j+1}.$

The desired conclusion now follows by Proposition 3.5. ∎

For example, consider the partition $\lambda=(3,3,1,1,0,\ldots)$ of $N=8$. The
exceptional degrees are

${K_{\lambda}}=\\{0,1,2,3,5,6,9,10\\}.$

The degree set is therefore,

${I_{\lambda}}=\\{4,7,8,11,12,13,\ldots\\},$

with $4,7,8$ being the sporadic degrees. In terms of the above terminology,

$\displaystyle K_{1}$ $\displaystyle=\\{0,1,2,3,4,5,6,9,10\\},$
$\displaystyle\lambda^{(1)}$ $\displaystyle=(2,2,0,0,0,\ldots),\quad
N_{1}=4,\;\ell_{1}=2$ $\displaystyle K_{2}$
$\displaystyle=\\{0,1,2,3,4,5,6,7,9,10\\},$ $\displaystyle\lambda^{(2)}$
$\displaystyle=(1,1,0,0,0,\ldots),\quad N_{2}=2,\;\ell_{2}=2$ $\displaystyle
K_{3}$ $\displaystyle=\\{0,1,2,3,4,5,6,7,8,9,10\\},$
$\displaystyle\lambda^{(3)}$ $\displaystyle=(0,0,0,0,0,\ldots),\quad
N_{3}=0,\;\ell_{3}=0.$

###  3.2. The norm identity.

In this section, we present and prove a certain algebraic identity that will
allow us to derive a formula for the norming constants of the exceptional
Hermite polynomials.

######  Proposition 3.9.

Let $m_{1},\ldots,m_{l},m\in{\mathbb{N}_{0}}$ be distinct non-negative
integers. Set

$\displaystyle\eta_{0}$ $\displaystyle=1$ $\displaystyle\xi_{0}$
$\displaystyle=H_{m}$ $\displaystyle\eta_{i}$
$\displaystyle=\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}}\right),\quad
i=1,\ldots,l;$ $\displaystyle\xi_{i}$
$\displaystyle=\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}},H_{m}\right)$

Also, let $\rho_{0}=0$, and recursively define

$\rho_{i+1}=\frac{\xi_{i}\xi_{i+1}}{\eta_{i}\eta_{i+1}}+2(m-m_{i+1})\rho_{i},\quad
i=0,\ldots,l-1.$

We then have,

(60)
$\left(\frac{\xi_{l}}{\eta_{l}}\right)^{2}e^{-x^{2}}-2^{l}\prod_{i=1}^{l}(m-m_{i})\xi_{0}^{2}e^{-x^{2}}=\left(\rho_{l}\,e^{-x^{2}}\right)^{\prime}.$

###### Proof.

By (49), we have

$\chi(\eta_{i},\eta_{i+1})=2(i-m_{i+1})\eta_{i}\eta_{i+1},\quad
i=0,\ldots,l-1.$

Hence, by the argument used in Proposition 3.5, the differential expressions

$\displaystyle\alpha_{i}$ $\displaystyle=\alpha_{\eta_{i+1},\eta_{i}},\quad
i=0,\ldots,l-1$ $\displaystyle\beta_{i}$
$\displaystyle=\beta_{\eta_{i},\eta_{i+1}},\quad i=0,\ldots,l-1$
$\displaystyle\tau_{i}$ $\displaystyle=\tau_{\eta_{i}},\quad i=0,\ldots,l.$

constitute the factorization chain

$\displaystyle\tau_{i}$ $\displaystyle=\beta_{i}\alpha_{i}+2(i-m_{i+1}),\quad
i=0,\ldots,l-1$ $\displaystyle\tau_{i+1}$
$\displaystyle=\alpha_{i}\beta_{i}+2(i+1-m_{i+1}).$

Next, observe that

$\displaystyle\alpha_{i}\xi_{i}$
$\displaystyle=\frac{\operatorname{Wr}\\!\left(\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}},H_{m_{i+1}}\right),\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}},H_{m}\right)\right)}{\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}}\right)}$
$\displaystyle=\operatorname{Wr}\\!\left(H_{m_{1}},\ldots,H_{m_{i}},H_{m_{i+1}},H_{m}\right)=\xi_{i+1}.$

Again, by (49), we have

$\tau_{i}(\xi_{i})=2(i-m)\xi_{i},\quad i=0,\ldots,l.$

Hence, by (13), we have

$\displaystyle\frac{\xi_{i+1}^{2}}{\eta_{i+1}^{2}}e^{-x^{2}}-2(m-m_{i+1})\frac{\xi_{i}^{2}}{\eta_{i}^{2}}e^{-x^{2}}$
$\displaystyle=\left(\frac{\xi_{i}\xi_{i+1}}{\eta_{i}\eta_{i+1}}e^{-x^{2}}\right)^{\prime},\quad
i=0,\ldots,l-1.$

Hence, by induction,

$\displaystyle\frac{\xi_{l}^{2}}{\eta_{l}^{2}}e^{-x^{2}}-2^{l}\prod_{i=1}^{l}(m-m_{i})\xi_{0}^{2}e^{-x^{2}}=\left(e^{-x^{2}}\sum_{i=1}^{l}\prod_{j=1+i}^{l}(2m-2m_{j})\frac{\xi_{j}\xi_{j-1}}{\eta_{j}\eta_{j-1}}\right)^{\prime}$

Again, by induction,

$\rho_{k}=\sum_{i=1}^{k}\prod_{j=1+i}^{l}(2m-2m_{j})\frac{\xi_{j}\xi_{j-1}}{\eta_{j}\eta_{j-1}},\quad
k=1,\ldots,\ell-1.$

The desired identity (60) follows immediately. ∎

##  4\. The $\operatorname{\mathrm{L}}^{2}$ theory

The following theorem was proved by Krein for Sturm-Liouville problems on the
half-line, and independently by Adler for Sturm-Liouville problems on a
bounded interval. The Adler argument can be extended without difficulty to the
case of an infinite interval [7].

######  Definition 4.1.

We say that $\lambda$ is an even partition $\lambda_{2i-1}=\lambda_{2i}$ for
every $i\in\mathbb{N}$.

######  Theorem 4.1 (Krein-Adler).

The polynomial $\eta_{\lambda}(x)$ has no real zeros if and only if $\lambda$
is an even partition.

Henceforth we assume that $\lambda$ is an even partition. In that case,

$W_{\lambda}:=W_{\eta_{\lambda}}=\eta_{\lambda}^{-2}e^{-x^{2}}$

is a non-singular weight on $(-\infty,\infty)$ with finite moments. Let
$\mathcal{H}_{\lambda}=\operatorname{\mathrm{L}}^{2}(\mathbb{R},W_{\lambda})$
denote the corresponding Hilbert space with weighted inner product,

$\left\langle
f,g\right\rangle_{\lambda}=\int_{\mathbb{R}}f\bar{g}W_{\lambda}=\int_{R}\frac{f}{\eta_{\lambda}}\frac{\bar{g}}{\eta_{\lambda}}e^{-x^{2}},\quad
f,g\in\mathcal{H}_{\lambda}.$

######  Proposition 4.1.

Let $\lambda$ be an even partition. Then, the corresponding exceptional
Hermite polynomials enjoy the following orthogonality property:

(61) $\left\langle
H_{\lambda,n},H_{\lambda,n^{\prime}}\right\rangle_{\lambda}=\sqrt{\pi}\,2^{n-N}\frac{n!}{\pi_{\lambda,N}(n)}\,\delta_{n,n^{\prime}},\quad
n,n^{\prime}\in{I_{\lambda}},$

with $\pi_{\lambda,N}(n)$ as defined in (34).

###### Proof.

Being the eigenfunctions of a SLP (4), the $H_{\lambda,n},\;n\in{I_{\lambda}}$
are orthogonal with respect to $W_{\lambda}=W_{\eta_{\lambda}}$. It remains to
establish the form of the norming constants in (61). By Corollary 3.1 and
(60),

$\displaystyle\left\langle H_{\lambda,n},H_{\lambda,n}\right\rangle_{\lambda}$
$\displaystyle=\int_{\mathbb{R}}\left(\frac{H_{\lambda,n}(x)}{\eta_{\lambda}(x)}\right)^{2}e^{-x^{2}}dx,\quad
n\in{I_{\lambda}}$
$\displaystyle=\int_{\mathbb{R}}\left(\frac{\operatorname{Wr}\\!\left(H_{k_{N}},\ldots,H_{k_{1}},H_{n}\right)}{2^{N}\pi_{\lambda,N}(n)\operatorname{Wr}\\!\left(H_{k_{N}},\ldots,H_{k_{1}}\right)}\right)^{2}e^{-x^{2}}dx,$
$\displaystyle=\frac{1}{2^{N}\pi_{\lambda,N}(n)}\int_{R}H_{n}(x)^{2}e^{-x^{2}}dx$
$\displaystyle=\sqrt{\pi}2^{n-N}\frac{n!}{2^{N}\pi_{\lambda,N}(n)}.$

∎

######  Theorem 4.2.

Let $\lambda$ be an even partition. Then the SLP of form (5) with
$\eta=\eta_{\lambda}$ has eigenvalues

$\varepsilon_{n}=2(n-N),\quad n\in{I_{\lambda}}$

The corresponding eigenfunctions are $H_{\lambda,n},\;n\in{I_{\lambda}}$,
while the norming constants are given by (61).

######  Lemma 4.1.

Let $\lambda$ be a partition and let $\lambda^{(j)},\;j\in\mathbb{N}$ be the
partitions defined in (56). If $\lambda$ is an even partition, then so is
every $\lambda^{(j)}$.

###### Proof.

Suppose that $\lambda_{2i-1}=\lambda_{2i}$ for all $i\in\mathbb{N}$. It
follows that

$\lambda^{(j)}_{2i-1}=\max(\lambda_{2i-1}-j,0)=\max(\lambda_{2i}-j,0)=\lambda^{(j)}_{2i}\,\quad
i,j\in\mathbb{N}.$

∎

###### of Theorem 4.2.

Multiplication of (4) by $-W_{\lambda}^{-1}$ and changing
$\varepsilon\to-\varepsilon$, gives (3), which may be expressed as

${\tau_{\lambda}}y=\varepsilon y.$

By (12) and (52), we have

${\tau_{\lambda}}H_{\lambda,n}=2(N-n)H_{\lambda,n},\quad n\in{I_{\lambda}}.$

Hence, $2(n-N)$ is an eigenvalue of (5) with corresponding eigenfunction
$H_{\lambda,n},\;n\in{I_{\lambda}}$.

It remains to show that there are no either eigenvalues; i.e., that
$\\{H_{\lambda,n}\\}_{n\in{I_{\lambda}}}$ is a complete orthogonal basis. To
show this, we employ the factorization chain described in Proposition 3.8. Let
$\lambda^{(j)},\;j=0,\ldots,\lambda_{1}$ be the partitions defined in (56),
and let $\eta_{j}=\eta_{\lambda^{(j)}}$. By the above Lemma, these are all
even partitions, and hence each $\eta_{j}\in\mathcal{R}$.

Let $T_{j}=T_{\eta_{j}},\;j=0,1,\ldots,\lambda_{1}$ be a sequence of self-
adjoint operators with action $\tau_{\eta_{j}}$ and domains as per (20). By
Proposition 2.6, the formal factorization chain of Proposition 3.8 corresponds
to a factorization chain of operators

$\displaystyle T_{j}$ $\displaystyle=B_{j}A_{j}+2\ell_{j},\quad
j=0,\ldots,\lambda_{1}-1,$ $\displaystyle T_{j+1}$
$\displaystyle=A_{j}B_{j}+2\ell_{j}+2,\quad j=0,\ldots,\lambda_{1}-1,$

where $\ell_{j}$ is the length of partition $\lambda^{(j)}$, and where
$A_{j},B_{j}$ are first order operators with action
$\alpha_{\eta_{j+1},\eta_{j}},\beta_{\eta_{j+1},\eta_{j}}$, respectively, and
domains as given in (18). Hence, $2\ell_{j}$ is the largest eigenvalue of
$T_{j},\;j=0,\ldots,\lambda_{1}$. Hence, by Theorem 2.1,

$\sigma(T_{j+1}-2)=\sigma(T_{j})\setminus\\{2\ell_{j}\\}.$

It follows that

$\sigma(T_{0})=\\{2\ell_{j}-2j\colon
j=0,\ldots,\lambda_{1}-1\\}\cup\sigma(T_{\lambda_{1}}).$

However, since $\tau_{\lambda_{1}}=\tau_{\emptyset}$ is the classical Hermite
operator, we have by (57) that

$\displaystyle\sigma(T_{0})$ $\displaystyle=\\{2\ell_{j}-2j\colon
j=0,\ldots,\lambda_{1}-1\\}\cup(-2{\mathbb{N}_{0}})$
$\displaystyle=\\{2(N-n_{j+1})\colon
j=0,\ldots,\lambda_{1}-1\\}\cup(-2{\mathbb{N}_{0}})$
$\displaystyle=\\{2(N-n)\colon n\in{I_{\lambda}}\\}.$

∎

######  Corollary 4.1.

Let $K=\\{k_{1},\ldots,k_{N}\\}\subset{\mathbb{N}_{0}}$ be a list of $N$ non-
negative integers such that

$\sum_{i=1}^{N}k_{i}=\frac{1}{2}N(N+1).$

Set

$\lambda_{i}=k_{i}+i-N,\;i\in\mathbb{N}.$

Then, $\\{H_{\lambda,n}\\}_{n\in{I_{\lambda}}}$ is a family of exceptional
Hermite polynomials such that $K={K_{\lambda}}$; i.e. the exceptional degrees
are precisely $K$. Moreover, if $\lambda$ is an even partition, then the
corresponding $\\{H_{\lambda,n}\\}_{n\in{I_{\lambda}}}$ are eigenfunctions of
the SLP (4) whose spectrum, up to a shift with $N$, differs from the spectrum
of the classical Hermite SLP by the removal of eigenvalues at position
$2k,\;k\in K$.

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