Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity

Bispectrality of $\boldsymbol{N}$-Component KP Wave Functions:
A Study in Non-Commutativity

Alex KASMAN

A. Kasman

Department of Mathematics, College of Charleston, USA<EMAIL_ADDRESS>http://kasmana.people.cofc.edu

Received May 13, 2015, in final form October 28, 2015; Published online
November 01, 2015

A wave function of the $N$-component KP Hierarchy with continuous flows
determined by an invertible matrix $H$ is constructed from the choice of an
$MN$-dimensional space of finitely-supported vector distributions. This wave
function is shown to be an eigenfunction for a ring of matrix differential
operators in $x$ having eigenvalues that are matrix functions of the spectral
parameter $z$. If the space of distributions is invariant under left
multiplication by $H$, then a matrix coefficient differential-translation
operator in $z$ is shown to share this eigenfunction and have an eigenvalue
that is a matrix function of $x$. This paper not only generates new examples
of bispectral operators, it also explores the consequences of non-
commutativity for techniques and objects used in previous investigations.

bispectrality; multi-component KP hierarchy; Darboux transformations; non-
commutative solitons

34L05; 16S32; 37K10

##  1 Introduction

The “bispectral problem” seeks to identify linear operators $L$ acting on
functions of the variable $x$ and $\Lambda$ acting on functions of the
variable $z$ such that there exists an eigenfunction $\psi(x,z)$ satisfying
the equations

$\displaystyle L\psi=p(z)\psi\qquad\mbox{and}\qquad\Lambda\psi=\pi(x)\psi.$

In other words, the components of the bispectral triple
$(L,\Lambda,\psi(x,z))$ satisfy two different eigenvalue equations, but with
the roles of the spacial and spectral variables exchanged.

The search for bispectral triples was originally formulated and investigated
by Duistermaat and Grünbaum [8] in a paper which completely resolved the
question in the special case in which the operators were scalar differential
operators with one being a Schrödinger operator. Since then, the bispectrality
of many different sorts of operators have been considered and many connections
to different areas of math and physics have also been discovered. (See [20]
and the articles referenced therein.)

The present paper will be considering a type of bispectrality in which both
the operators and eigenvalues behave differently when acting from the left
than from the right. It is necessary to introduce some notation and reorder
the terms in the eigenvalue equations in order to properly describe the main
results.

Throughout the paper, $M$ and $N$ should be considered to be fixed (but
arbitrary) natural numbers. In addition, $H$ is a fixed (but arbitrary)
invertible constant $N\times N$ matrix. Let $\Delta_{n,\lambda}$ denote the
linear functional acting from the right on functions of $z$ by differentiating
$n$ times and evaluating at $z=\lambda$:

$\displaystyle(f(z))\Delta_{n,\lambda}=f^{(n)}(\lambda).$

The set of linear combinations of these finitely-supported distributions with
coefficients from $\mathbb{C}^{N}$ will be denoted $\mathbb{D}$:

$\displaystyle\mathbb{D}=\left\\{\sum_{j=1}^{m}\Delta_{n_{j},\lambda_{j}}C_{j}\colon
m,(n_{j}+1)\in\mathbb{N},\ \lambda_{j}\in\mathbb{C},\
C_{j}\in\mathbb{C}^{N}\right\\}.$

Finally, let $W^{*}\subset\mathbb{D}$ be an $MN$-dimensional space of
distributions.

The goal of this paper is to produce from the selection of $H$ and $W^{*}$ a
bispectral triple $(L,\Lambda,\psi)$ such that

  * •

$\psi(x,z)=(I+O(z^{-1}))e^{xzH}$ is an $N\times N$ matrix function of $x$ and
$z$ with the specified asymptotics in $z$,

  * •

$\psi(x,z)(zH)^{M}$ is holomorphic in $z$ and in the kernel of every element
of $W^{*}$ (i.e., $\psi$ “satisfies the conditions”),

  * •

$L\psi(x,z)=\psi(x,z)p(z)$ for the matrix differential operator in $x$ acting
from the left and some matrix function $p(z)$,

  * •

and $\psi(x,z)\Lambda=\pi(x)\psi(x,z)$ for a matrix differential-translation
operator in $z$ acting from the right and some matrix function $\pi(x)$.

In the case $N=1$, this goal is already achieved constructively for any choice
of distributions [23]. One interesting result of the present paper is that for
$N>1$, lack of commutativity with $H$ may impose an obstacle to finding such a
bispectral triple in that no such triple exists for certain choices of
$W^{*}$. The three subsections below each offer some motivation for interest
in the existence of such triples.

###  1.1 New bispectral triples

One source of interest in the present paper is simply the fact that it
generates examples of bispectral triples that have not previously been
studied. The construction outlined below produces many bispectral triples
$(L,\Lambda,\psi)$, where $L$ is a matrix coefficient differential operator in
$x$, $\Lambda$ is an operator in $z$ which acts by both differentiation and
translation in $z$, and $\psi(x,z)$ is a matrix function which asymptotically
approaches $e^{xzH}$ for a chosen invertible matrix $H$.

This can be seen either as a matrix generalization of the paper [23] which
considered exactly this sort of bispectrality in the scalar case or as a
generalization of the matrix bispectrality in [2, 4, 32] to the a more general
class of eigenfunctions and operators.

###  1.2 Bispectral duality of integrable particle systems

Among the applications found for bispectrality is its surprising role in the
duality of integrable particle systems. Two integrable Hamiltonian systems are
said to be “dual” in the sense of Ruijsenaars if the action-angle maps
linearizing one system is simply the inverse of the action-angle map of the
other [26]. When the Hamiltonians of the systems are quantized, the
Hamiltonian operators themselves share a common eigenfunction and form a
bispectral triple [11].

Moreover, the duality of the classical particle systems can also be manifested
through bispectrality in that the dynamics of the two operators in a
bispectral triple under some integrable hierarchy can be seen to display the
particle motion of the two dual systems respectively. This classical
bispectral duality was observed first in the case of the self-duality of the
Calogero–Moser system [21, 31]. In [23], it was conjectured that certain
bispectral triples involving scalar operators that translate in $z$ would
similarly be related to the duality of the rational Ruijsenaars–Schneider and
hyperbolic Calogero–Moser particle systems. This was later confirmed by Haine
[19].

The spin generalization of the Calogero–Moser system similarly exhibits
classical bispectral duality [2, 32]. Achieving this result essentially
involved generalizing the scalar case [21, 31] to bispectrality for matrix
coefficient differential operators. It is hoped that the construction
presented in this paper which generalize that in [23] will similarly find
application to classical bispectral duality in some future matrix
generalization of the results in [19].

###  1.3 Non-commutative bispectral Darboux transformations

In the original context of operators on scalar functions, the eigenvalue
equations defining bispectrality were originally written with all operators
and eigenvalues acting from the left. However, research into non-commutative
versions of the discrete-continuous version of the bispectral problem in non-
commutative contexts found it necessary to have operators in different
variables acting from opposite sides in order to ensure that they commute [5,
14, 15, 16, 17, 18] (cf. [9]). Continuous-continuous bispectrality in the non-
commutative context is also a subject of interest [6, 13, 27, 33]. As in the
discrete case, it was found that the generalizing the results from the scalar
case to the matrix case required letting the operators act from opposite sites
[2, 4, 32]. Building on this observation, the present paper seeks to further
consider the influence of non-commutativity on constructions and results
already known for scalar bispectral triples.

In this regard, the wave functions of the $N$-component KP hierarchy are of
interest since they asymptotically look like matrices of the form $e^{xzH}$,
where $H$ is an $N\times N$ matrix [3, 7]. Consequently, unlike the scalar
case or the case $H=I$ considered in [2, 32], the vacuum eigenfunction itself
may not commute with the coefficients of the operators. In fact, for the
purpose of more fully investigating the consequences of non-commutativity for
bispectral Darboux transformations, this paper will go beyond the standard
formalism for the $N$-component KP hierarchy by considering the case in which
$H$ is not even diagonalizable and therefore has a centralizer with more
interesting structure. Furthermore, following the suggestion of Grünbaum [13],
the present paper will consider the case in which both of the eigenvalues are
matrix-valued.

By generalizing the construction from [23] to the context in which the vacuum
eigenfunction, eigenvalues, and operator coefficients all generally fail to
commute with each other, this investigation has identified some results that
are surprisingly different than in the commutative case. For example, it is
shown that in this context there exist rational Darboux transformations that
do not preserve the bispectrality of the eigenfunction (see Section 6.1) and
that bispectral triples do not always exhibit ad-nilpotency (see Remark 6.1).
These will be summarized in the last section of the paper.

##  2 Additional notation

###  2.1 Distributions and matrices

Let $M$, $N$, $H$ and $W^{*}$ be as in the Introduction. The set of constant
$N$-component column vectors will be denoted by $\mathbb{C}^{N}$ and
$\mathbb{C}_{N\times N}$ is the set of $N\times N$ constant matrices.
Associated to the selection of $H$ one has

$\displaystyle\mathcal{C}=\\{Q\in\mathbb{C}_{N\times N}\colon[Q,H]=0\\},$

the centralizer of $H$ in $\mathbb{C}_{N\times N}$.

Let $\\{\delta_{1},\ldots,\delta_{MN}\\}$ be a basis for
$W^{*}\subset\mathbb{D}$. Unlike the selection of $N$, $M$ and $H$ which were
indeed entirely arbitrary, two additional assumptions regarding the choice of
$W^{*}$ will have to be made so that a bispectral triple may be produced from
it. However, rather than making those assumptions here at the start, the
additional assumptions will be introduced only when they become necessary.
This should help to clarify which results are independent of and which rely on
the assumptions.

Nearly all of the objects and constructions below depend on the choice of the
number $N$, the matrix $H$ and the distributions $W^{*}$ that have been
selected and fixed above, but to avoid complicating the notation the
dependence on these selections will not be written explicitly. (For instance,
the matrix $\Phi$ in (3.1) could be called $\Phi_{N,M,H,W^{*}}$ because it
does depend on these selections, but it will simply be called $\Phi$.)

###  2.2 Operators and eigenvalues

The operators in $x$ to be considered in this paper will all be differential
operators in the variable $x$ (also sometimes called $t_{1}$) which are
polynomials in $\partial=\frac{\partial}{\partial x}$ having coefficients that
are $N\times N$ matrix functions of $x$. The operators in $z$ will be written
in terms of $\partial_{z}=\frac{\partial}{\partial z}$ or the translation
operator $\top_{\alpha}\colon f(z)\mapsto f(z+\alpha)$. More generally, they
will be polynomials in these having coefficients that are $N\times N$ matrix
rational functions of $z$.

Because operator coefficients and eigenvalues will be matrix-valued, the
action of an operator will depend on whether its coefficients multiply from
the right or the left. It also matters whether the eigenvalue acts by
multiplication on the right or the left of the eigenfunction. This paper will
adopt the convention that all operators in $x$ that are independent of $z$
(whether they are differential operators or simply functions acting by
multiplication) act from the left and that all operators in $z$ that are
independent of $x$ (including functions, translation operators, finitely-
supported distributions and differential operators) act from the right. The
action of an operator in $z$ will be denoted simply by writing the operator to
the right of the function it is acting on. So, for instance, the function
$e^{xzH}$ satisfies the eigenvalue equations

$\displaystyle\partial e^{xzH}=e^{xzH}(zH)\qquad\text{and}\qquad
e^{xzH}\partial_{z}=(xH)e^{xzH}.$

The decision to have operators in $x$ and $z$ acting from different sides is
not merely a matter of notation. The need for such an assumption for the
differential operators in $x$ and $z$ respectively was already noted in prior
work on matrix bispectrality [2, 4, 14, 15, 16, 17, 18, 32]. The present work
extends this convention to the eigenvalues and finitely-supported
distributions as well, and does so because the theorems fail to be true
otherwise.

######  Remark 2.1.

Note that one needs to be cautious about applying intuition about
eigenfunctions in a commutative setting without considering how non-
commutativity may affect it. For example, although a non-zero multiple of an
eigenfunction in the commutative setting always remains an eigenfunction with
the same eigenvalue, here there are two other possibilities. Suppose
$L\psi(x,z)=\psi(x,z)p(z)$, so that $\psi$ is an eigenfunction for $L$ with
eigenvalue $p$, and that $g$ is an invertible constant matrix. Then $g\psi$
may not be an eigenfunction for $L$ if $[L,g]\not=0$ and more surprisingly
even though $\psi g$ is an eigenfunction for $L$, the corresponding eigenvalue
changes to $g^{-1}pg$.

##  3 Dual construction for $\boldsymbol{N}$-component KP

The purpose of this section is to produce a matrix coefficient pseudo-
differential operator satisfying the Lax equations of the multicomponent KP
hierarchy introduced by Date–Jimbo–Kashiwara–Miwa [7] and mostly follows the
approach of Segal–Wilson [29]. The proof methods utilized here are rather
standard in the field of integrable systems. However, one of the main points
of this paper is that some of the novel features of this situation pose
unexpected obstacles to the standard methods used to study bispectrality. So,
although it is not surprising that the operators produced in this way satisfy
these Lax equations, the proofs are presented with sufficient detail to ensure
that they work despite the non-diagonalizability of $H$ and the fact that the
distributions here are acting from the right.

###  3.1 The $\boldsymbol{N}$-component Sato Grassmannian

Let $\mathcal{H}^{(N)}$ denote the Hilbert space of square-integrable vector-
valued functions $S^{1}\to(\mathbb{C}^{N})^{\top}$ where
$S^{1}\subset\mathbb{C}$ is the unit circle $|z|=1$ and
$(\mathbb{C}^{N})^{\top}$ is the set of complex valued row111In previous
papers the elements of $\mathcal{H}^{(N)}$ have been written as column
vectors. However, because the construction of bispectral operators below is
most easily described in terms of matrix finitely-supported distributions in
$z$ acting from the right, they will be written here as $1\times N$ matrices.
$N$-vectors. Denote by $e_{i}$ for $0\leq i\leq N-1$ the $1\times N$ matrix
which has the value $1$ in column $i+1$ and zero in the others. This extends
to a basis $\\{e_{i}\colon i\in\mathbb{Z}\\}$ of $\mathcal{H}^{(N)}$ for which
$e_{i}=z^{a}e_{b}$ when $i=aN+b$ for $0\leq b\leq N-1$. The Hilbert space has
the decomposition

$\displaystyle\mathcal{H}^{(N)}=\mathcal{H}^{(N)}_{+}\oplus\mathcal{H}^{(N)}_{-},$

where $\mathcal{H}^{(N)}_{+}$ is the Hilbert closure of the subspace spanned
by $e_{i}$ for $0\leq i$ and $\mathcal{H}^{(N)}_{-}$ is the Hilbert closure of
the subspace spanned by $e_{i}$ for $i<0$.

######  Definition 3.1.

The Grassmannian ${\rm Gr}^{(N)}$ is set of all closed subspaces
$V\subset\mathcal{H}^{(N)}$ such that the orthogonal projections
$V\to\mathcal{H}^{(N)}_{-}$ is a compact operator and such that the orthogonal
projection $V\to\mathcal{H}^{(N)}_{+}$ is Fredholm of index zero [3, 28, 29].

The notion of $N$-component KP hierarchy to be considered in this paper is
compatible with, but somewhat different from that addressed by previous
authors as the following remark explains.

######  Remark 3.2.

For the $N$-component KP hierarchy, the construction of solutions from a point
in the Grassmannian usually involves a collection of diagonal constant
matrices $H_{\alpha}$ ($1\leq\alpha\leq N$) such that powers of $zH_{\alpha}$
infinitesimally generate the continuous flows and $z$-dependent matrices
$T_{\beta}$ ($1\leq\beta\leq N-1$) that generate discrete flows (sometimes
called “Schlesinger transformations”) of the hierarchy [3, 7]. In the present
paper, however, only the continuous flows generated infinitesimally by powers
of $z$ times the (not necessarily diagonal) matrix $H$ selected earlier will
be considered.

###  3.2 A point of $\boldsymbol{{\rm Gr}^{(N)}}$ associated to the selection
of distributions

As usual, one associates a subspace of $\mathcal{H}^{(N)}$ to the choice of
$W^{*}$ by taking its dual in $\mathcal{H}^{(N)}_{+}$ and multiplying on the
right by the inverse of a matrix polynomial in $z$ whose degree depends on the
dimension of $W^{*}$ (cf. [21, 22, 29, 30] where the analogous procedure
involved dividing by a scalar polynomial):

######  Definition 3.3.

Let $W\subset\mathcal{H}^{(N)}$ be defined by

$\displaystyle W=\big{\\{}p(z)(zH)^{-M}\colon p(z)\in\mathcal{H}^{(N)}_{+},\
(p)\delta=0\ \text{for}\ \delta\in W^{*}\big{\\}}.$

######  Lemma 3.4.

$W\in{\rm Gr}^{(N)}$.

###### Proof.

The image of $W$ under the projection map onto $\mathcal{H}^{(N)}_{-}$ is
contained in the finite-dimensional subspace spanned by the basis elements
$e_{i}$ for $-MN\leq i\leq-1$. This is sufficient to conclude that the
projection map is compact. The map $w\mapsto w(zH)^{M}$ from $W$ to
$\mathcal{H}^{(N)}_{+}$ has Fredholm index $MN$ because it has no kernel and
the image is the common solution set of $MN$-linearly independent conditions.
The map from $\mathcal{H}^{(N)}_{+}$ which first right multiplies by
$(zH)^{-M}$ and then projects onto $\mathcal{H}^{(N)}_{+}$ has index $-MN$
since its kernel is spanned by the basis vectors $e_{i}$ with $0\leq i\leq
MN-1$ but the image is all of $\mathcal{H}^{(N)}_{+}$. The composition of
these maps is the projection from $W$ to $\mathcal{H}^{(N)}_{+}$ and so its
index is the sum of the indices which is zero.∎

###  3.3 $\boldsymbol{N}$-component KP wave function

######  Definition 3.5.

Let $\psi_{0}=\exp\big{(}\sum\limits_{i=1}^{\infty}t_{i}z^{i}H^{i}\big{)}$
where $\textbf{t}=(t_{1},t_{2},\ldots)$ are the continuous KP time variables,
with the variables $t_{1}$ and $x$ considered to be identical. Let
$\phi_{i}(\textbf{t})=(\psi_{0})\delta_{i}$ ($1\leq i\leq MN$) be the
$\mathbb{C}^{N}$-valued functions obtained by applying each element of the
basis of $W^{*}$ to $\psi_{0}$. Combine them as blocks into the $N\times MN$
matrix $\boldsymbol{\phi}=(\phi_{1}\ \cdots\ \phi_{MN})$ and define the matrix
$\Phi$ as the $MN\times MN$ block Wronskian matrix

$\displaystyle\Phi(\textbf{t})=\left(\begin{matrix}\boldsymbol{\phi}\\\
\dfrac{\partial}{\partial x}\boldsymbol{\phi}\\\ \vdots\\\
\dfrac{\partial^{M-1}}{\partial
x^{M-1}}\boldsymbol{\phi}\end{matrix}\right)=\left(\begin{matrix}\phi_{1}&\phi_{2}&\cdots&\phi_{MN}\\\
\phi_{1}^{\prime}&\phi_{2}^{\prime}&\cdots&\phi_{MN}^{\prime}\\\
\vdots&\vdots&\ddots&\vdots&\\\
\phi_{1}^{(M-1)}&\phi_{2}^{(M-1)}&\cdots&\phi_{MN}^{(M-1)}\end{matrix}\right).$
(3.1)

######  Assumption 3.6.

Henceforth, assume that $W^{*}$ was chosen so that the matrix $\Phi$ in (3.1)
is invertible for some values of $x=t_{1}$ (i.e., so that $\det\Phi\not\equiv
0$).

The following remarks offer two different interpretations of the fact that
Assumption 3.6 is necessary here but not for the analogous result in the
scalar case [23].

######  Remark 3.7.

When $N=1$, the requirement that $\det(\Phi)\not=0$ is equivalent to the
requirement that $\\{\phi_{1},\ldots,\phi_{M}\\}$ is a linearly independent
set of functions. Then, the independence of the basis of distributions would
already ensure that Assumption 3.6 is satisfied. However, when $N>1$ the block
Wronskian matrix $\Phi$ can be singular even if the functions $\phi_{i}$ are
linearly independent as functions of $x$. (For example, consider the case
$M=1$, $N=2$, $\phi_{1}=(1\ 1)^{\top}$, $\phi_{2}=(x\ x)^{\top}$).

######  Remark 3.8.

The determinant of $\Phi$ can be interpreted as the determinant of the
projection map from $\psi_{0}^{-1}(\textbf{t})W$ to $\mathcal{H}^{(N)}_{+}$.
In other words, it is the $\tau$-function of $W$. (The proof of this claim is
essentially the same as the proof of Theorem 7.5 in [22].) The $\tau$-function
is non-zero when $\psi_{0}^{-1}(\textbf{t})W$ is in the “big cell” of the
Grassmannian. In the case $N=1$, the orbit of any point $W$ in the
Grassmannian under the action of $\psi_{0}^{-1}$ intersects the big cell [29].
In contrast, for the $N$-component KP hierarchy it is known that there are
points in the Grassmannian whose orbit under the continuous flows never
intersect the big cell [3].

######  Definition 3.9.

Due to Assumption 3.6, we may define the differential operator $K$ as

$K=\partial^{M}I-\left(\phi_{1}^{(M)}\ \cdots\
\phi_{MN}^{(M)}\right)\Phi^{-1}\left(\begin{matrix}I\\\ \partial I\\\
\vdots\\\ \partial^{M-1}I\end{matrix}\right).$ (3.2)

######  Lemma 3.10.

  1. $(a)$

The operator $K$ defined in (3.2) is the unique monic $N\times N$ differential
operator of order $M$ such that222The expression $D\boldsymbol{\phi}=0$ is a
convenient way to write that applying the $N\times N$ differential operator
$D$ to each function $\phi_{i}$ ($1\leq i\leq MN$) results in the zero vector
of $\mathbb{C}^{N}$. $K\boldsymbol{\phi}=0$.

  2. $(b)$

If $L$ is any $N\times N$ matrix differential operator satisfying
$L\boldsymbol{\phi}=0$ then $L=Q\circ K$ for some differential operator $Q$.

It is easy to check that $K\boldsymbol{\phi}=0$. Alternatively, Lemma 3.10(a)
follows from results of Etingof, Gelfand and Retakh on quasi-determinants
[10]. However, Lemma 3.10(b) is apparently a new result. Although the lemma
was originally formulated for this paper333Lemma 3.10(b) is used in the proofs
of Theorem 3.15, Lemma 5.10 and Theorem 4.7., a self-contained proof is being
published separately as [24].

######  Remark 3.11.

The theory of quasi-determinants is not explicitly being used here but the
operator $K$ defined in (3.2) could alternatively be computed as a quasi-
determinant of a Wronskian matrix with $N\times N$ matrix entries [10]. The
method of quasi-determinants was applied to the bispectral problem for matrix
coefficient operators in [4]. So, in this sense, the use of this operator $K$
here is a continuation of the approach adopted there.

######  Definition 3.12.

Let $\psi(\textbf{t},z)=K(\psi_{0})(zH)^{-M}.$ This function will play an
important role as the eigenfunction for the operators in $x$ and (given one
additional assumption) $z$ to be introduced below.

######  Theorem 3.13.

The function $\psi$ defined above has the following properties:

  * •

$\psi(\textbf{t},z)=(I+O(z^{-1}))\psi_{0}$ where $I$ is the $N\times N$
identity matrix,

  * •

$\psi(\textbf{t},z)\in W$ for all t in the domain of $\psi$.

Consequently, $\psi=\psi_{W}$ is the $N$-component KP wave function of the
point $W\in{\rm Gr}^{(N)}$.

###### Proof.

Because $\partial(\psi_{0})=\psi_{0}zH$, applying the monic differential
operator $K=\partial^{M}+\cdots$ to $\psi_{0}$ produces a function of the form
$K\psi_{0}=P(\textbf{t},z)\psi_{0}$ where $P$ is a polynomial of degree $M$ in
$z$ with leading coefficient $H^{M}$. Then
$K\psi_{0}H^{-M}z^{-M}=(I+O(z^{-1}))\psi_{0}$ as claimed. It remains to be
shown that $\psi$ is an element of $W$. By Definition 3.3 it is sufficient to
note that for each $1\leq i\leq MN$, $\psi z^{M}H^{M}=K\psi_{0}$ satisfies

$\displaystyle(K\psi_{0})\delta_{i}=K((\psi_{0})\delta_{i})=K\phi_{i}=0.$ ∎

###  3.4 Lax equations of the $\boldsymbol{N}$-component KP hierarchy

######  Definition 3.14.

Let $K^{-1}$ denote the unique multiplicative inverse of the monic
differential operator $K$ in the ring of matrix-coefficient pseudo-
differential operators and let $\mathcal{L}=K\circ\partial\circ K^{-1}$ be the
pseudo-differential operator obtained by conjugating $\partial$ by $K$.

######  Theorem 3.15.

$\mathcal{L}$ satisfies the Lax equations

$\displaystyle\frac{\partial}{\partial
t_{i}}\mathcal{L}=[(\mathcal{L}^{i})_{+},\mathcal{L}]$

for each $i\in\mathbb{N}$, where
$\big{(}\sum\limits_{i=-\infty}^{n}\alpha_{i}(\textbf{t})\partial^{i}\big{)}_{+}=\sum\limits_{i=0}^{n}\alpha_{i}(\textbf{t})\partial^{i}$.

###### Proof.

Let $\phi=(\psi_{0})\delta$ for some $\delta\in W^{*}$. A key observation is
that for each $i\in\mathbb{N}$:

$\displaystyle\frac{\partial}{\partial t_{i}}\phi=\frac{\partial}{\partial
t_{i}}(\psi_{0})\delta=\left(\frac{\partial}{\partial
t_{i}}\psi_{0}\right)\delta=\left((zH)^{i}\psi_{0}\right)\delta=\left(\frac{\partial^{i}}{\partial
x^{i}}\psi_{0}\right)\delta=\partial^{i}(\psi_{0})\delta=\frac{\partial^{i}}{\partial
x^{i}}(\phi).$

Using the fact that $\phi$ satisfies these “dispersion relations” and the
intertwining relationship $\mathcal{L}^{i}\circ K=K\circ\partial^{i}$ we
differentiate the identity $K(\phi)=0$ (which follows from Lemma 3.10) by
$t_{i}$ to get

$\displaystyle
0=K_{t_{i}}(\phi)+K(\phi_{t_{i}})=K_{t_{i}}(\phi)+K(\partial^{i}\phi)=K_{t_{i}}(\phi)+\mathcal{L}^{i}\circ
K(\phi)$ $\displaystyle\hphantom{0}=K_{t_{i}}(\phi)+(\mathcal{L}^{i})_{-}\circ
K(\phi)+(\mathcal{L}^{i})_{+}\circ K(\phi).$

Since both $K$ and $(\mathcal{L}^{i})_{+}$ are ordinary differential operators
(as “$+$” denotes the projection onto the subring of ordinary differential
operators), $(\mathcal{L}^{i})_{+}\circ K$ is an ordinary differential
operator with a right factor of $K$. Therefore, $\phi$ is in its kernel and
the last term in the sum above is zero. We may therefore conclude that

$\displaystyle K_{t_{i}}(\phi)+(\mathcal{L}^{i})_{-}\circ K(\phi)=0.$ (3.3)

(Note that the second term in this sum need not be zero since
$(\mathcal{L}^{i})_{-}$ is not an ordinary differential operator.)

Using $\mathcal{L}^{i}\circ K=K\circ\partial^{i}$ we can split
$\mathcal{L}^{i}$ into its positive and negative parts to get

$\displaystyle(\mathcal{L}^{i})_{-}\circ
K=K\circ\partial^{i}-(\mathcal{L}^{i})_{+}\circ K.$

Since the object on the right is just a difference of differential operators
we know that $(\mathcal{L}^{i})_{-}\circ K$ is a differential operator.

According to (3.3), $\Gamma(\boldsymbol{\phi})=0$ where $\Gamma$ is the
ordinary differential operator

$\displaystyle\Gamma=K_{t_{i}}+(\mathcal{L}^{i})_{-}\circ K.$

Then by Lemma 3.10, there exists a differential operator $Q$ so that
$\Gamma=Q\circ K$. However, $\Gamma$ has order strictly less than $M$ since
the coefficient of the $M^{\rm th}$ order term of $K$ is constant by
construction and since multiplying by $(\mathcal{L}^{i})_{-}$ will necessarily
lower the order. This is only possible if $Q=0$ and $\Gamma$ is the zero
operator. Hence, $K_{t_{i}}=-(\mathcal{L}^{i})_{-}\circ K$. The Lax equation
follows because

$\displaystyle\mathcal{L}_{t_{i}}=K_{t_{i}}\circ\partial\circ
K^{-1}-K\circ\partial\circ K^{-1}\circ K_{t_{i}}\circ K^{-1}$
$\displaystyle\hphantom{\mathcal{L}_{t_{i}}}{}=-(\mathcal{L}^{i})_{-}\circ
K\circ\partial\circ K^{-1}+K\circ\partial\circ
K^{-1}\circ(\mathcal{L}^{i})_{-}\circ K\circ K^{-1}$
$\displaystyle\hphantom{\mathcal{L}_{t_{i}}}{}=[\mathcal{L},(\mathcal{L}^{i})_{-}]=[(\mathcal{L}^{i})_{+},\mathcal{L}].$
∎

##  4 Operators in $\boldsymbol{x}$ having $\boldsymbol{\psi}$ as
eigenfunction

######  Remark 4.1.

From this point onwards, the goal is to determine whether the wave function
$\psi(\textbf{t},z)$ is an eigenfunction for an operator in $x=t_{1}$ with
$z$-dependent eigenvalue and vice versa. The higher indexed time variables
will only complicate the notation. So, it will henceforth be assumed that
$t_{i}=0$ for $i\geq 2$. Then, $\boldsymbol{\phi}$ and the coefficients of $K$
can be considered to be functions of only the variable $x$, and
$\psi_{0}(x,z)=e^{xzH}$ and $\psi(x,z)$ (sometimes called the “stationary wave
function”) are functions of $x$ and $z$. Unlike the numbered assumptions, this
one is made for notational simplicity only. Dependence on the KP time
variables can be added to the objects to be discussed below so that all claims
remain valid.

######  Definition 4.2.

A distribution $\delta\in\mathbb{D}$ can be composed with
$p(z)\in\mathbb{C}_{N\times N}[z]$ by defining $p(z)\circ\delta$ to be the
distribution whose value when applied to $f(z)$ is the same as that of
$\delta$ applied to the product $f(z)p(z)$ for any $f(z)\in\mathbb{C}_{N\times
N}[z]$. Associate to the choice $W^{*}$ of distributions the ring of
polynomials with coefficients in $\mathcal{C}$ which turn elements of $W^{*}$
into elements of $W^{*}$:

$\displaystyle\mathcal{A}=\big{\\{}p(z)\in\mathcal{C}[z]\colon
p(z)\circ\delta\in W^{*}\ \forall\,\delta\in W^{*}\big{\\}}.$

In particular, for each $p\in\mathcal{A}$ and each basis element $\delta_{i}$
there exist numbers $c_{j}$ such that
$p\circ\delta_{i}=\sum\limits_{i=1}^{MN}\delta_{j}c_{j}$.

As in the scalar case, elements of $\mathcal{A}$ are stabilizers of the point
in the Grassmannian: if $p\in\mathcal{A}$ then $Wp\subset W$. The main
significance of the ring $\mathcal{A}$ is that there is a differential
operator $L_{p}$ of positive order satisfying $L_{p}\psi=\psi p(z)$ for every
non-constant $p\in\mathcal{A}$. Interestingly, unlike the scalar case, it will
be shown that the question of which constant matrices are in $\mathcal{A}$ is
also of interest in that whether $\psi$ is part of a bispectral triple is
related to whether $H$ is an element of $\mathcal{A}$. Before those facts are
established, however, the following definitions and results show that
$\mathcal{A}$ contains polynomials of every sufficiently high degree.

######  Definition 4.3.

Let $S\subset\mathbb{C}$ denote the support of the distributions in $W^{*}$.
That is, $\lambda\in S$ if an only if $\Delta_{n,\lambda}$ appears with non-
zero coefficient for some $n$ in at least one $\delta_{i}$. For each
$\lambda\in S$ let $m_{\lambda}$ denote the largest number $n$ such that
$\Delta_{n,\lambda}$ appears with non-zero coefficient in at least one element
of $W^{*}$. The scalar polynomial

$\displaystyle p_{0}(z)=\prod_{\lambda\in S}(z-\lambda)^{m_{\lambda}+1}$

will be used in the next lemma, in Definition 5.8 and also in Theorem 5.13
below.

######  Lemma 4.4.

For any $p\in\mathcal{C}[z]$, the product $p_{0}(z)p(z)$ is in $\mathcal{A}$.
So, $p_{0}(z)\mathcal{C}[z]\subset\mathcal{A}.$

###### Proof.

Let $p\in\mathcal{C}[z]$ and $\delta\in W^{*}$. Then applying the distribution
$p_{0}p\circ\delta$ to any polynomial $q$ is equal to $(qp_{0}p)\delta$. The
distribution $\delta$ will act by differentiating and evaluating at
$z=\lambda$ for each $\lambda\in S$. However, $p_{0}$ was constructed so that
it has zeroes of sufficiently high multiplicity at each $\lambda$ to ensure
that this will be equal to zero. Hence $p_{0}p\circ\delta$ is the zero
distribution, which trivially satisfies the criterion in the definition of
$\mathcal{A}$. ∎

######  Definition 4.5.

For $p\in\mathcal{C}[z]$ where $p=\sum\limits_{i=0}^{n}C_{i}z^{i}$ for
$C_{i}\in\mathcal{C}$ define

$\displaystyle\flat^{-1}(p)=\sum_{i=0}^{n}C_{i}H^{-i}\partial^{i}.$

######  Lemma 4.6.

For any $p\in\mathcal{C}[z]$, the constant coefficient differential operator
$\flat^{-1}(p)$ has $\psi_{0}=e^{xzH}$ as an eigenfunction with eigenvalue
$p(z)$.

###### Proof.

If $p$ is the polynomial with coefficients $C_{i}$ as in Definition 4.5 then

$\displaystyle\flat^{-1}(p)\psi_{0}=\left(\sum_{i=0}^{n}C_{i}H^{-i}\partial^{i}\right)\psi_{0}=\sum_{i=0}^{n}C_{i}H^{-i}\partial^{i}\psi_{0}=\sum_{i=0}^{n}C_{i}H^{-i}\psi_{0}(zH)^{i}$
$\displaystyle\hphantom{\flat^{-1}(p)\psi_{0}}{}=\psi_{0}\left(\sum_{i=0}^{n}C_{i}H^{-i}(zH)^{i}\right)=\psi_{0}\left(\sum_{i=0}^{n}C_{i}z^{i}\right)=\psi_{0}p(z).$
∎

######  Theorem 4.7.

For any $p\in\mathcal{A}$ there is an $N\times N$ ordinary differential
operator $L_{p}$ in $x$ satisfying the intertwining relationship

$\displaystyle L_{p}\circ K=K\circ\flat^{-1}(p)$

and the eigenvalue equation $L_{p}\psi(x,z)=\psi p(z)$.

###### Proof.

Let $p\in\mathcal{A}$. Observe that

$\displaystyle
K\circ\flat^{-1}(p)(\phi_{i})=K\circ\flat^{-1}(p)((\psi_{0})\delta_{i})=K\big{(}\big{(}\flat^{-1}(p)\psi_{0}\big{)}\delta_{i}\big{)}=K((\psi_{0})p(z)\circ\delta_{i})$
$\displaystyle\hphantom{K\circ\flat^{-1}(p)(\phi_{i})}{}=K\left(\sum_{j=1}^{MN}(\psi_{0})\delta_{j}c_{j}\right)=\sum_{j=1}^{MN}K(\phi_{j})c_{j}=\sum_{j=1}^{MN}0\times
c_{j}=0.$

But, that means that each function $\phi_{i}$ is in the kernel of
$K\circ\flat^{-1}(p)$ and hence by Lemma 3.10 there is a differential operator
$L_{p}$ such that $K\circ\flat^{-1}(p)=L_{p}\circ K$. This establishes the
intertwining relationship.

Applying $L_{p}$ to $\psi=K\psi_{0}(zH)^{-M}$ one finds

$\displaystyle L_{p}\psi=L_{p}\big{(}K\psi_{0}(zH)^{-M}\big{)}=(L_{p}\circ
K)\psi_{0}(zH)^{-M}=\big{(}K\circ\flat^{-1}(p)\big{)}\psi_{0}(zH)^{-M}$
$\displaystyle\hphantom{L_{p}\psi}{}=K\psi_{0}p(z)(zH)^{-M}=K\psi_{0}(zH)^{-M}p(z)=\psi
p(z).$ ∎

##  5 Operators in $\boldsymbol{z}$ having $\boldsymbol{\psi}$ as
eigenfunction

######  Definition 5.1.

For any $\alpha\in\mathbb{C}$ let the translation operator $\top_{\alpha}$ act
on functions of $z$ according to the definition

$\displaystyle(f(z))\top_{\alpha}=f(z+\alpha).$

Further let
$\top_{\alpha}^{H}=\sum_{i,j}\top_{\alpha\gamma_{i}}(\alpha\partial_{z})^{j}C_{ij}$
where the matrices $C_{ij}$ and constants $\gamma_{i}$ are defined by the
formula

$\displaystyle\exp\big{(}{xH^{-1}}\big{)}=\sum_{i,j}\exp({\gamma_{i}x})x^{j}C_{ij}\qquad\text{(with
$\gamma_{i}\not=\gamma_{i^{\prime}}$ if $i\not=i^{\prime}$).}$ (5.1)

######  Lemma 5.2.

The differential-translation operator $\top_{\alpha}^{H}$ has
$\psi_{0}=e^{xzH}$ as an eigenfunction with eigenvalue $e^{\alpha x}$:

$\displaystyle\psi_{0}\top_{\alpha}^{H}=e^{\alpha x}\psi_{0}.$

###### Proof.

By definition,

$\displaystyle
e^{xzH}\top_{\alpha}^{H}=e^{xzH}\sum_{i,j}T_{\alpha\gamma_{i}}(\alpha\partial_{z}^{j})C_{ij}=e^{xzH}\sum_{i,j}e^{x\alpha\gamma_{i}H}(\alpha
xH)^{j}C_{ij}.$

Then the claim follows if it can be shown that the sum in the last expression
is equal to $e^{\alpha x}$.

However, (5.1) holds not only for any scalar $x$ but also when it is replaced
by some matrix which commutes with the matrices $H^{-1}$ and $C_{ij}$ which
appear in it. In particular, since $\alpha xH$ commutes with $H^{-1}$, its
commutator with the expression on the left side of equation (5.1) is zero.
From this one can determine that its commutator with each coefficient matrix
$C_{ij}$ on the right is zero as well. However, replacing $x$ with $\alpha xH$
yields the formula

$\displaystyle\exp(\alpha x)=\sum_{i,j}\exp({\gamma_{i}\alpha xH})(\alpha
xH)^{j}C_{ij}$

as needed. ∎

The goal of this section is to produce operators in $z$ which are matrix
differential-translation operators in $z$ having rational coefficients that
share the eigenfunction $\psi$ with the differential operators $L_{p}$ in $x$
constructed in the previous section. In order for the construction to work, an
additional assumption is required:

######  Assumption 5.3.

Henceforth, it is assumed that $H\in\mathcal{A}$. Equivalently, assume that
for each $1\leq i\leq MN$ there exist numbers $c_{j}$ such that

$\displaystyle H\delta_{i}=\sum_{j=1}^{MN}c_{j}\delta_{j}.$

###  5.1 The anti-isomorphism

This section will introduce an anti-isomorphism between rings of operators in
$x$ and $z$ respectively such that an operator and its image have the same
action on $\psi_{0}$. The use of such an anti-isomorphism as a method for
studying bispectrality was pioneered in special cases in [30] and [25] and
extended to a very general commutative context in [1]. (Additionally, three
months after the present paper was posted and submitted, a new preprint by two
of the same authors as [1] appeared which seeks to further generalize those
results to the non-commutative context [12].)

######  Definition 5.4.

The two rings of operators of interest to this construction are

$\displaystyle\mathcal{W}=\bigoplus_{\alpha\in\mathbb{C}}e^{\alpha
x}\mathcal{C}[x,\partial]\qquad\hbox{and}\qquad\mathcal{W}^{\flat}=\bigoplus_{\alpha\in\mathbb{C}}\top_{\alpha}^{H}\mathcal{C}[z,\partial].$

Note that operators from both rings involve polynomial coefficient matrix
differential operators which commute with the constant matrix $H$, but
elements of $\mathcal{W}$ may also include a finite number of factors of the
form $e^{\alpha x}$ while elements of $\mathcal{W}^{\flat}$ may similarly
include factors of $\top_{\alpha}^{H}$ for a finite number of complex numbers
$\alpha$.

######  Definition 5.5.

Let $\flat\colon\mathcal{W}\to\mathcal{W}^{\flat}$ be defined by

$\displaystyle\flat\left(\sum_{i=0}^{l}\sum_{j=0}^{m}\sum_{k=1}^{n}C_{ijk}e^{\alpha_{k}x}x^{i}\partial^{j}\right)=\sum_{i=0}^{l}\sum_{j=0}^{m}\sum_{k=1}^{n}\partial_{z}^{i}\top_{\alpha_{k}}^{H}z^{j}C_{ijk}H^{j-i},$
(5.2)

where $C_{ijk}\in\mathcal{C}$ are the coefficient matrices.

######  Lemma 5.6.

For any $L_{0}\in\mathcal{W}$, the operators $L_{0}$ and $\flat(L_{0})$ have
the same action on $\psi_{0}=e^{xzH}$:

$\displaystyle L_{0}\psi_{0}=\psi_{0}\flat(L_{0}).$

Moreover, $\flat$ is an anti-isomorphism.

###### Proof.

Since

$\displaystyle\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}x^{i}e^{\alpha_{k}x}\partial^{j}(\psi_{0})=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}e^{\alpha_{k}x}x^{i}(zH)^{j}(\psi_{0})$
$\displaystyle\qquad{}=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}e^{\alpha_{k}x}x^{i}(\psi_{0})(zH)^{j}=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}e^{\alpha_{k}x}(\psi_{0})\big{(}H^{-1}\partial_{z}\big{)}^{i}(zH)^{j}$
$\displaystyle\qquad{}=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}(\psi_{0})\top_{\alpha_{k}}^{H}\big{(}H^{-1}\partial_{z}\big{)}^{i}(zH)^{j}=\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}(\psi_{0})\big{(}H^{-1}\partial_{z}\big{)}^{i}\top_{\alpha_{k}}^{H}(zH)^{j}C_{ij}$
$\displaystyle\qquad{}=(\psi_{0})\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}\big{(}H^{-1}\partial_{z}\big{)}^{i}\top_{\alpha_{k}}^{H}(zH)^{j}C_{ij}=\psi_{0}\flat\left(\sum_{i=0}^{m}\sum_{j=0}^{n}\sum_{k}C_{ij}x^{i}e^{\alpha_{k}x}\partial^{j}\right)$

it follows that $L_{0}$ and $\flat(L_{0})$ have the same action on $\psi_{0}$.

The inverse map $\flat^{-1}$ is found by simply moving the factor of $H^{j-i}$
to the other side of equation (5.2), which confirms that $\flat$ is a
bijection. If $K_{1},K_{2}\in\mathcal{W}$ we have

$\displaystyle(K_{2}\circ
K_{1})\psi_{0}=K_{2}(K_{1}(\psi_{0}))=K_{2}(\psi_{0}\flat(K_{1}))=(K_{2}(\psi_{0}))\flat(K_{1})$
$\displaystyle\hphantom{(K_{2}\circ
K_{1})\psi_{0}}{}=(\psi_{0}\flat(K_{2}))\flat(K_{1})=\psi_{0}(\flat(K_{2})\circ\flat(K_{1})).$

However, we also know that $(K_{2}\circ
K_{1})\psi_{0}=\psi_{0}\flat(K_{2}\circ K_{1})$. Then, $\flat(K_{2}\circ
K_{1})-\flat(K_{2})\circ\flat(K_{1})$ has the $\psi_{0}$ in its kernel. The
only operator in $\mathcal{W}^{\flat}$ having $\psi_{0}$ in its kernel is the
zero operator. Therefore,

$\displaystyle\flat(K_{2}\circ K_{1})=\flat(K_{2})\circ\flat(K_{1})$ (5.3)

and the map is an anti-isomorphism. ∎

######  Remark 5.7.

The reader may be surprised to see “anti-isomorphism” being used to describe a
map satisfying equation (5.3). Generally, a map having this property is called
an isomorphism. There are two reasons this terminology is being used here.
First, that is the terminology that was used for the analogous map in previous
papers in the scalar setting [1, 23, 25]. More importantly, the fact that the
order of operators here is preserved is merely a consequence of the fact that
the operators in $z$ act from the right while the operators in $x$ act from
the left. Note that when $K_{2}\circ K_{1}$ acts on a function it is $K_{1}$
that acts first while when $\flat(K_{2})\circ\flat(K_{1})$ acts it is
$\flat(K_{2})$ that acts first and it is in this sense that it is an anti-
isomorphism. Nevertheless, it is interesting to note that in this context the
map does preserve the order of operators in a product. It may be that had the
operators in different variables been written as acting from opposite sides
even in the commutative case from the beginning, the map would have been
described as an isomorphism instead.

###  5.2 Eigenvalue equations for operators in $\boldsymbol{z}$

######  Definition 5.8.

Let $L_{0}$ be the constant coefficient matrix differential operator

$\displaystyle L_{0}=\flat^{-1}(p_{0}(z)I).$

######  Lemma 5.9.

$L_{0}\boldsymbol{\phi}=0$ and so $L_{0}=Q\circ K$ for some $Q$.

###### Proof.

Because differentiation in $x$ and multiplication on the left commute with the
application of the distributions $\delta_{i}$ we have

$\displaystyle
L_{0}\phi_{i}=L_{0}(\psi_{0})\delta_{i}=(L_{0}\psi_{0})\delta_{i}.$

By Lemma 5.6, $L_{0}\psi_{0}=\psi_{0}p_{0}(z)$. The polynomial $p_{0}$ was
chosen so that it has a zero of high enough multiplicity at each point in the
support of $\delta_{i}$ to guarantee that

$\displaystyle(\psi_{0}p_{0}(z))\delta_{i}=0.$

It then follows from Lemma 3.10 that $L_{0}=Q\circ K$. ∎

In the remainder of this construction, $Q$ will denote the differential
operator such that $L_{0}=Q\circ K$.

######  Lemma 5.10.

Given Assumption 5.3, the operators $L_{0}$, $K$ and $Q$ commute with $H$.

###### Proof.

Applying the operator $H^{-1}KH$ to $\phi_{i}$ we see that

$\displaystyle
H^{-1}KH\phi_{i}=H^{-1}KH\big{(}e^{xzH}\big{)}\delta_{i}=H^{-1}K\big{(}e^{xzH}\big{)}H\delta_{i}=H^{-1}K\left(\sum_{j=1}^{MN}c_{j}\phi_{j}\right)$
$\displaystyle\hphantom{H^{-1}KH\phi_{i}}{}=H^{-1}\left(\sum_{j=1}^{MN}c_{j}K(\phi_{j})\right)=0.$

However, by Lemma 3.10, $K$ is the unique monic operator of order $M$ having
all the basis functions $\phi_{i}$ in its kernel so $H^{-1}KH=K$.

We also know that $L_{0}$ commutes with $H$ because $p_{0}$ is a scalar
multiple of the identity and $\flat^{-1}$ which turns $p_{0}$ into $L_{0}$
only introduces additional powers of $H$. Then

$\displaystyle HQ\circ K=HL_{0}=L_{0}H=Q\circ KH=Q\circ HK=QH\circ K.$

Multiplying this by $K^{-1}$ on the right yields $HQ=QH$. ∎

Note that $K$ and $Q$ are probably not in $\mathcal{W}$, as they may be
rational functions in $x$ and a finite number of functions of the form
$e^{\alpha x}$. However, it is possible to clear their denominators either by
multiplying by a function on the left or by composing with a function on the
right, which motivates the following definition:

######  Definition 5.11.

Let $\mathcal{A}^{\flat}$ be the set of $x$-dependent $N\times N$ matrix
functions defined as follows:

$\displaystyle\mathcal{A}^{\flat}=\left\\{\hat{\pi}\in\bigoplus_{\alpha\in\mathbb{C}}\mathcal{C}[x]e^{\alpha
x}\colon\hat{\pi}=\pi_{Q}\pi_{K},\
\pi_{K}(x)K\in\mathcal{W},Q\circ\pi_{K}(x)\in\mathcal{W}\right\\}.$

In other words, it is the set of zero order elements of $\mathcal{W}$ which
factor as a product such the right factor times $K$ and $Q$ composed with the
left factor are both elements of $\mathcal{W}$.

######  Lemma 5.12.

$\mathcal{A}^{\flat}$ is non-empty and contains matrix functions that are non-
constant in $x$.

###### Proof.

Note that $K$ and $Q$ are operators that are rational in $x$ and a finite
number of terms of the form $e^{x}$. If we let $\pi_{K}$ be the least common
multiple of the denominators of the coefficients in $K$ then $\pi_{K}K$ is in
$\mathcal{W}$ (since we have simply cleared the denominator by
multiplication). Similarly, if we let $\pi_{Q}$ be a high enough power of the
least common multiple of the denominators of $Q$ then
$Q\circ\pi_{Q}\in\mathcal{W}$. Then, $\hat{\pi}=\pi_{Q}\pi_{K}$ is by
construction an element of $\mathcal{A}^{\flat}$. Since, $\pi_{Q}f\pi_{K}$ is
also an element of $\mathcal{A}^{\flat}$ for any order zero element of
$\mathcal{W}$, $\mathcal{A}^{\flat}$ contains non-constant matrix functions. ∎

The main result is the construction of an operator $\Lambda$ in $z$ with
eigenvalue $\hat{\pi}$ when applied to the wave function $\psi$ for every
$\hat{\pi}\in\mathcal{A}^{\flat}$:

######  Theorem 5.13.

Let $\hat{\pi}=\pi_{Q}\pi_{K}\in\mathcal{A}^{\flat}$ where
$\bar{K}=\pi_{K}K\in\mathcal{W}$ and $\bar{Q}=Q\circ\pi_{Q}\in\mathcal{W}$.
Define
$\Lambda:=(zH)^{M}\circ(p_{0}(z))^{-1}\circ\flat(\bar{Q})\circ\flat(\bar{K})\circ(zH)^{-M}$,
then

$\displaystyle\psi\Lambda=\hat{\pi}(x)\psi.$

###### Proof.

Write $L_{0}$ as $L_{0}=\bar{Q}\circ(\pi_{K}(x)\pi_{Q}(x))^{-1}\circ\bar{K}$.
Applying this operator to $\psi_{0}$ and multiplying each side by
$p_{0}^{-1}(z)$ gives

$\displaystyle(\bar{Q}\circ(\pi_{K}(x)\pi_{Q}(x))^{-1}\circ\bar{K})\psi_{0}p_{0}^{-1}(z)=\psi_{0}.$

Moving $\bar{K}$ to the other side using the anti-isomorphism and applying
$\flat(\bar{Q})$ to both sides this becomes

$\displaystyle\big{(}\bar{Q}\circ(\pi_{K}(x)\pi_{Q}(x))^{-1}\big{)}\psi_{0}(\flat(\bar{K})\circ
p_{0}^{-1}(z)\circ\flat(\bar{Q}))=\psi_{0}\flat(\bar{Q}).$

Moving the last expression to the other side of the equality and moving
$\flat(\bar{Q})$ in it to the other side, we finally get

$\displaystyle\bar{Q}((\pi_{K}\pi_{Q})^{-1}\psi_{0}\big{(}\flat(\bar{K})\circ
p_{0}^{-1}\circ\flat(\bar{Q})\big{)}-\psi_{0})=0.$

Note that $\bar{Q}$ is a differential operator in $x$ with a non-singular
leading coefficient (since it is a factor of the monic differential operator
$L_{0}$). Hence, its kernel is finite-dimensional. The only way the expression
to which it is applied, a polynomial in $z$ multiplied by $e^{xzH}$, could be
in its kernel for all $z$ is if it is equal to zero. From this, we conclude
that

$\displaystyle\flat(\bar{K})\circ
p_{0}^{-1}\circ\flat(\bar{Q})=\flat(\pi_{K}\pi_{Q}).$

Using this one finds that the action of $\Lambda$ on $\psi$ is

$\displaystyle(\psi)\Lambda=\big{(}K\psi_{0}(zH)^{-M}\big{)}\Lambda=\big{(}\pi_{K}^{-1}(x)\bar{K}\psi_{0}(zH)^{-M}\big{)}\Lambda$
$\displaystyle\hphantom{(\psi)\Lambda}{}=\big{(}\pi_{K}^{-1}(x)(\psi_{0})\flat{\bar{K}}(zH)^{-M}\big{)}\Lambda=\big{(}\pi_{K}^{-1}(x)\psi_{0}\big{)}\flat{\bar{K}}\circ(zH)^{-M}\circ\Lambda$
$\displaystyle\hphantom{(\psi)\Lambda}{}=\big{(}\pi_{K}^{-1}(x)\psi_{0}\big{)}\flat{\bar{K}}\circ(p_{0}(z))^{-1}\circ\flat(\bar{Q})\circ\flat(\bar{K})\circ(zH)^{-M}$
$\displaystyle\hphantom{(\psi)\Lambda}{}=\big{(}\pi_{K}^{-1}(x)\psi_{0}\big{)}\Lambda_{0}\circ\flat(\bar{K})\circ(zH)^{-M}=\big{(}\pi_{K}^{-1}(x)\pi_{K}(x)\pi_{Q}(x)\psi_{0}\big{)}\circ\flat(\bar{K})\circ(zH)^{-M}$
$\displaystyle\hphantom{(\psi)\Lambda}{}=(\pi_{Q}(x)\psi_{0})\circ\flat(\bar{K})\circ(zH)^{-M}=\big{(}\pi_{Q}(x)\pi_{K}(x)\pi_{K}^{-1}(x)\psi_{0}\big{)}\circ\flat(\bar{K})\circ(zH)^{-M}$
$\displaystyle\hphantom{(\psi)\Lambda}{}=\hat{\pi}(x)(\pi_{K}^{-1}(x)\psi_{0})\flat(\bar{K})\circ(zH)^{-M}=\hat{\pi}(x)\big{(}\pi_{K}^{-1}(x)\bar{K}\psi_{0}\big{)}(zH)^{-M}$
$\displaystyle\hphantom{(\psi)\Lambda}{}=\hat{\pi}(x)(K\psi_{0})(zH)^{-M}=\hat{\pi}(x)\psi.$
∎

###  5.3 Ad-nilpotency

We now have eigenvalue equations $L_{p}\psi=\psi p(z)$ and
$\psi\Lambda=\hat{\pi}(x)\psi$. In the commutative case, Duistermaat and
Grünbaum noticed that these operators and eigenvalues satisfied an interesting
relationship that goes by the name of “ad-nilpotency” [8]. As usual, for
operators $R$ and $P$ we recursively define $\textup{ad}_{R}^{n}P$ by

$\displaystyle\textup{ad}_{R}^{1}P=R\circ P-P\circ R\ \hbox{and}\
\textup{ad}_{R}^{n}P=R\circ\big{(}\textup{ad}_{R}^{n-1}P\big{)}-\big{(}\textup{ad}_{R}^{n-1}P\big{)}\circ
R,\qquad n\geq 2.$

When $R$ is a differential operator of order greater than one, then one
generally expects the order of $\textup{ad}_{R}^{n}P$ to get large as $n$ goes
to infinity, but they found that when $R$ is a scalar bispectral differential
operator and $P$ is the eigenvalue of a corresponding differential operator in
the spectral variable, then surprisingly $\textup{ad}_{R}^{n}P$ is the zero
operator for large enough $n$. In fact, in that scalar Schrödinger operator
case they considered, Duistermaat and Grünbaum found that ad-nilpotency was
both a necessary and sufficient condition for bispectrality [8].

As it turns out, in the this new context even the easier of these two
statements fails to hold. In particular, the ad-nilpotency in the commutative
case is a consequence of the fact that the leading coefficients of $R\circ P$
and $P\circ R$ are equal, but this is not generally true for differential
operators with matrix coefficients. Consequently, it is not the case that ad-
nilpotency holds for all of the bispectral operators produced by the procedure
described above. The following results (and an example in Remark 6.1) show how
and to what extent the old result generalizes.

######  Theorem 5.14.

Given $L_{p}$, $\Lambda$, $\hat{\pi}(x)$ and $p(z)$ as above, the results of
applying $\textup{ad}_{\hat{\pi}}^{n}L_{p}$ and $\textup{ad}_{\Lambda}^{n}p$
on $\psi$ are equal for every $n\in\mathbb{N}$.

###### Proof.

The case $n=1$ is obtained by subtracting the equations

$\displaystyle\hat{\pi}(x)L_{p}\psi=\hat{\pi}(x)\psi p(z)=\psi\Lambda p(z)$

from the equations with the orders of the operators reversed

$\displaystyle L_{p}\hat{\pi}(x)\psi=L_{p}\psi\Lambda=\psi p(z)\Lambda.$

If one assumes the claim is true for $n=k$ then the case $n=k+1$ is proved
similarly and the general case follows by induction. ∎

So, the equivalence of the actions of the two operators generated by iterating
“ad” remains true regardless of non-commutativity. One cannot conclude from
this alone that either of them is zero without making further assumptions, but
if $\textup{ad}_{\hat{\pi}}^{n}L=0$ (which would be the case, for instance, if
one could be certain that $\hat{\pi}$ would commute with the coefficients)
then because $\psi$ is not in the kernel of any non-zero operator in $z$
alone, the one could conclude the same is true for the corresponding operator
written in terms of $\Lambda$ and $p$:

######  Corollary 5.15.

If $\textup{ad}_{\hat{\pi}}^{n}L=0$ then $\textup{ad}_{\Lambda}^{n}p=0$.

######  Remark 5.16.

This may be the first time that ad-nilpotency has been considered for
translation operators as well as for matrix operators. It is therefore
relevant to note that Corollary 5.15 is valid even when the operator $\Lambda$
involves shift operators of the form $\top_{\alpha}$ (see Remark 6.1).

##  6 Examples

###  6.1 A wave function that is not part of a bispectral triple

Consider the case
$W^{*}=\operatorname{span}\left\\{\delta_{1},\delta_{2}\right\\}$,

$\displaystyle H=\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right),\qquad\delta_{1}=\left(\begin{matrix}\Delta_{1,0}\\\
\Delta_{0,0}\end{matrix}\right),\qquad\text{and}\qquad\delta_{2}=\left(\begin{matrix}0\\\
\Delta_{1,0}\end{matrix}\right).$

Then Assumption 3.6 is met and

$\displaystyle\psi(x,z)=\left(\begin{matrix}\dfrac{xz-1}{xz}e^{xz}&0\vspace{1mm}\\\
\dfrac{1}{x^{2}z}e^{xz}&\dfrac{xz+1}{xz}e^{-xz}\end{matrix}\right).$

The matrix polynomial $p(z)=z^{2}I$ satisfies $p\circ\delta_{i}=0$ for $i=1$
and $i=2$ and so $p\in\mathcal{A}$ and as predicted by Theorem 4.7, the
corresponding operator

$\displaystyle
L_{p}=\left(\begin{matrix}\partial^{2}-\dfrac{2}{x^{2}}&0\vspace{1mm}\\\
\dfrac{4}{x^{3}}&\partial^{2}-\dfrac{2}{x^{2}}\end{matrix}\right)$

satisfies $L_{p}\psi=\psi p$.

Note that this is a rational Darboux transformation of the first operator and
eigenfunction from the bispectral triple
$(\partial^{2},\partial_{z}^{2},e^{xzH})$. Since in the scalar case it has
been found that rational Darboux transformations preserve bispectrality, one
might expect $L_{p}$ and $\psi$ to be part of a bispectral triple. However,
$H\delta_{i}\not\in W^{*}$ and so Assumption 5.3 is not satisfied. Thus,
Theorem 5.13 does not guarantee the existence of a differential-translation
operator $\Lambda$ in $z$ having $\psi$ as an eigenfunction. In fact, in this
simple case we can see that no such operator exists.

Briefly, the argument is as follows. Consider a differential-translation
operator $\Lambda$ acting on $\psi$ from the right and suppose there is a
function $\pi(x)$ such that $\psi\Lambda=\pi(x)\psi$. By noting the
coefficients of $e^{xz}$ and $e^{-xz}$ in the top right entry of each side of
this equality, we see immediately that $\Lambda_{12}=\pi_{12}=0$ (i.e.,
$\Lambda$ and $\pi$ are both lower triangular.). Then,we have the scalar
eigenvalue equation $\psi_{11}\Lambda_{11}=\pi_{11}\psi_{11}$ and also
$\psi_{21}\Lambda_{11}=\pi_{21}\psi_{11}+\pi_{22}\psi_{21}$. Combining these
with $\psi_{11}+x\psi_{21}=e^{xz}$ one concludes that $e^{xz}\Lambda_{11}$ can
be written in the form $(f(x)+g(x)/z)e^{xz}$. It follows that $\Lambda_{11}$
as an operator in $z$ has only coefficients that are constant or are a
constant multiplied by $1/z$. In fact, all of the operators in $z$ having
$\psi_{11}$ as eigenfunction are known; they are the operators that intertwine
by $\partial-\frac{1}{z}$ with a constant coefficient operator. The only ones
that meet both the criteria of the previous two sentences are the order zero
operators. In other words, $\Lambda_{11}$ would have to be a number. Then the
equation for the action of $\Lambda_{11}$ on $\psi_{21}$ tells us that
$\pi_{21}=0$ and $\pi_{22}=\pi_{11}$ is also a number. Since the eigenvalue of
this operator $\Lambda$ does not depend on $x$, the operator does not form a
bispectral triple with $L$ and $\psi$.

###  6.2 A rational example with non-diagonalizable $\boldsymbol{H}$

Consider $H=I+U$, where

$\displaystyle U=\left(\begin{matrix}0&1\\\
0&0\end{matrix}\right),\qquad\text{so
that}\qquad\psi_{0}=e^{xzH}=\left(\begin{matrix}e^{xz}&ze^{xz}\\\
0&e^{xz}\end{matrix}\right).$

The distributions chosen are
$W^{*}=\operatorname{span}\\{\delta_{1},\dots,\delta_{4}\\}$ with

$\displaystyle\delta_{1}=\left(\begin{matrix}\Delta_{2,1}\\\
0\end{matrix}\right),\qquad\delta_{2}=\left(\begin{matrix}0\\\
\Delta_{2,1}\end{matrix}\right),\qquad\delta_{3}=\left(\begin{matrix}\Delta_{0,1}\\\
0\end{matrix}\right),\qquad\text{and}\qquad\delta_{4}=\left(\begin{matrix}0\\\
\Delta_{0,1}\end{matrix}\right).$

Assumption 3.6 is met and the unique monic operator of order $2$ satisfying
$K(\phi_{i})=0$ where $\phi_{i}=(\psi_{0})\delta_{i}$ for $1\leq i\leq 4$ is

$\displaystyle K=\partial^{2}I+\left(\begin{matrix}\dfrac{-2x-1}{x}&-2\\\
0&\dfrac{-2x-1}{x}\end{matrix}\right)\partial+\left(\begin{matrix}\dfrac{x+1}{x}&\dfrac{2x^{2}+x}{x^{2}}\vspace{1mm}\\\
0&\dfrac{x+1}{x}\end{matrix}\right).$

Now define

$\displaystyle\psi=K(\psi_{0})(zH)^{-2}=\left(\begin{matrix}\dfrac{xz^{2}-2xz-z+x+1}{xz^{2}}&\dfrac{z-1}{xz^{2}}\vspace{1mm}\\\
0&\dfrac{xz^{2}-2xz-z+x+1}{xz^{2}}\end{matrix}\right)e^{xzH}.$

Assumption 5.3 is met because $H\delta_{i}=\delta_{i}$,
$H\delta_{i+1}=\delta_{i}+\delta_{i+1}$ for $i=1,3$. Consequently, it will be
possible to produce operators in $z$ sharing the eigenfunction $\psi$. The
matrix polynomial

$\displaystyle p(z)=\left(\begin{matrix}(z-1)^{2}&(z-1)^{3}\\\
0&(z-1)^{2}\end{matrix}\right)=(z-1)^{2}I+(z-1)^{3}U$

is in $\mathcal{A}$ because $p\circ\delta_{i}=2\delta_{i+2}$ and
$p\circ\delta_{i+2}=0$ for $i=1,2$. So,

$\displaystyle\flat^{-1}(p(z))=\left(\begin{matrix}\partial^{2}-2\partial+1&\partial^{3}-5\partial^{2}+5\partial-1\\\
0&\partial^{3}-2\partial+1\end{matrix}\right)$

satisfies the intertwining relationship

$\displaystyle K\circ\flat^{-1}(p(z))=L_{p}\circ K$

with

$\displaystyle L_{p}=\left(\begin{matrix}0&1\cr
0&0\end{matrix}\right)\partial^{3}+\left(\begin{matrix}1&-5\\\
0&1\end{matrix}\right)\partial^{2}+\left(\begin{matrix}-2&5-\dfrac{3}{x^{2}}\vspace{1mm}\\\
0&-2\end{matrix}\right)\partial+\left(\begin{matrix}1-\dfrac{2}{x^{2}}&-1+\dfrac{7}{x^{2}}+\dfrac{3}{x^{3}}\vspace{1mm}\\\
0&1-\dfrac{2}{x^{2}}\end{matrix}\right).$

This operator has eigenfunction $\psi$ with eigenvalue $p$:

$\displaystyle L_{p}\psi=L_{p}\circ K\psi_{0}=K\circ
L_{0}\psi_{0}=K\psi_{0}p(z)=\psi p(z).$

(As mentioned in Remark 2.1, multiplying $\psi$ by a matrix on the right has
the effect of conjugating the eigenvalue. It is therefore worth noting that
the eigenvalue $p$ here is not only non-diagonal but in fact non-
diagonalizable.)

To produce an operator in $z$ sharing $\psi$ as eigenfunction, we consider the
polynomial $p_{0}(z)=(z-1)^{3}$ (which has this form because $\lambda=1$ is
the only point in the support of the distributions in $W^{*}$ and because the
highest derivative they take there is $m_{\lambda}=2$). Then

$\displaystyle
L_{0}=\flat^{-1}(p_{0}(z)I)=\big{(}\partial^{3}-3\partial^{2}+3\partial-1\big{)}I+\big{(}{-}3\partial^{3}+6\partial^{2}-3\partial\big{)}U$

is the operator satisfying $L_{0}\psi_{0}=\psi_{0}p_{0}(z)$ and $L_{0}$
factors as $L_{0}=Q\circ K$ with

$\displaystyle
Q=\left(\begin{matrix}\partial+\dfrac{1-x}{x}&-3\partial+\dfrac{2x-3}{x}\vspace{1mm}\\\
0&\partial+\dfrac{1-x}{x}\end{matrix}\right).$

The next step is to choose two functions from $\mathcal{C}[x]$, so that
$\bar{K}=\pi_{K}K$ and $\bar{Q}=Q\circ\pi_{Q}$ are in
$\mathbb{C}[x,\partial]$. It turns out that the selection

$\displaystyle\pi_{K}=\left(\begin{matrix}x&x^{2}\\\
0&x\end{matrix}\right)=\pi_{Q}$

works and we get that

$\displaystyle\Lambda=(zH)^{2}\circ(p_{0}(z))^{-1}\circ\flat(\bar{Q})\circ\flat(\bar{K})\circ(zH)^{-2}=\partial_{z}^{3}I+\partial_{z}^{2}\left(\begin{matrix}1&-\dfrac{2\left(z^{2}-z+6\right)}{(z-1)z}\\\
0&1\end{matrix}\right)$
$\displaystyle\hphantom{\Lambda=}{}+\partial_{z}\left(\begin{matrix}\dfrac{4}{z-z^{2}}&\dfrac{2\left(z^{2}+8z+6\right)}{(z-1)^{2}z^{2}}\vspace{1mm}\\\
0&\dfrac{4}{z-z^{2}}\end{matrix}\right)+\left(\begin{matrix}\dfrac{-2z^{2}+4z+2}{(z-1)^{2}z^{2}}&\dfrac{4z^{3}-6z^{2}-8z-20}{(z-1)^{3}z^{2}}\vspace{1mm}\\\
0&\dfrac{-2z^{2}+4z+2}{(z-1)^{2}z^{2}}\end{matrix}\right)$

does indeed satisfy $\psi\Lambda=\pi_{Q}(x)\pi_{K}(x)\psi$.

###  6.3 Computing $\boldsymbol{\top_{\alpha}^{H}}$ for non-diagonal
$\boldsymbol{H}$

Suppose $H=\lambda I+U$ where $U$ is still the upper-triangular matrix from
the previous example. Since

$\displaystyle\hbox{exp}\big{(}xH^{-1}\big{)}=e^{x/\lambda}I-\frac{x}{\lambda^{2}}e^{x/\lambda}U,$

the corresponding operator formed by replacing $e^{x/\lambda}$ with
$\top_{\alpha/\lambda}$ and other occurrences of $x$ with $\alpha\partial_{z}$
would be

$\displaystyle\top_{\alpha}^{H}=\top_{\alpha/\lambda}I-\frac{1}{\lambda^{2}}\top_{\alpha/\lambda}\alpha\partial_{z}U.$

This operator in $z$ that combines differentiation and translation has the
property that $(e^{xzH})\top_{\alpha}^{H}$ $=e^{\alpha x}e^{xzH}$.

###  6.4 An exponential example with $\boldsymbol{H=I}$

Finally, consider $H=I$ and

$\displaystyle
W^{*}=\operatorname{span}\left\\{\left(\begin{matrix}\Delta_{0,0}+\Delta_{0,1}\\\
\Delta_{0,1}\end{matrix}\right),\left(\begin{matrix}0\\\
-\Delta_{0,0}+\Delta_{0,1}\end{matrix}\right)\right\\}.$

Then

$\displaystyle
K=\left(\begin{matrix}\partial-\dfrac{e^{x}}{1+e^{x}}&0\vspace{1mm}\\\
\dfrac{e^{x}}{e^{2x}-1}&\partial+\dfrac{e^{x}}{1-e^{x}}\end{matrix}\right)\qquad\text{and}$
$\displaystyle\psi(x,z)=\left(\begin{matrix}\dfrac{e^{xz}\left(e^{x}(z-1)+z\right)}{\left(1+e^{x}\right)z}&0\vspace{1mm}\\\
\dfrac{e^{zx+x}}{\left(-1+e^{2x}\right)z}&\dfrac{e^{xz}\left(e^{x}(z-1)-z\right)}{\left(-1+e^{x}\right)z}\end{matrix}\right).$

Because of the support of the distributions, $p_{0}(z)=z^{2}-z$ and we may
choose $p\in\mathcal{A}$ to be a multiple of that by a constant matrix

$\displaystyle p(z)=\left(\begin{matrix}0&z^{2}-z\\\
z^{2}-z&0\end{matrix}\right).$

The constant operator $\flat^{-1}(p)=p(\partial)$, satisfies the intertwining
relationship $K\circ p(\partial)=L_{p}\circ K$ with

$\displaystyle
L_{p}=\left(\begin{matrix}-\dfrac{e^{x}\left(-1+e^{x}-2e^{2x}\right)}{\left(-1+e^{2x}\right)^{2}}&-\dfrac{2e^{2x}}{\left(-1+e^{x}\right)^{2}\left(1+e^{x}\right)}\vspace{1mm}\\\
\dfrac{e^{2x}\left(-3+2e^{x}\right)}{\left(-1+e^{2x}\right)^{2}}&\dfrac{e^{x}}{\left(-1+e^{x}\right)^{2}\left(1+e^{x}\right)}\end{matrix}\right)$
$\displaystyle\hphantom{L_{p}=}{}+\left(\begin{matrix}-\dfrac{e^{x}}{-1+e^{2x}}&\dfrac{-1-e^{x}+3e^{2x}-e^{3x}}{\left(-1+e^{x}\right)^{2}\left(1+e^{x}\right)}\vspace{1mm}\\\
\dfrac{-1+2e^{x}+2e^{2x}-2e^{3x}-e^{4x}}{\left(-1+e^{2x}\right)^{2}}&\dfrac{e^{x}}{\left(-1+e^{x}\right)\left(1+e^{x}\right)}\end{matrix}\right)\partial+\left(\begin{matrix}0&1\\\
1&0\\\ \end{matrix}\right)\partial_{x}^{2}.$

Hence $L_{p}\psi=\psi p(z).$ (Since $p(z)$ and $\psi$ do not commute, it is
necessary to write the eigenvalue on the right rather than the left:
$L\psi\not=p(z)\psi$.)

Now, to compute an operator in $z$ having the same function $\psi$ as an
eigenfunction, we factor $p_{0}(\partial)I$ as $Q\circ K$ to obtain

$\displaystyle Q=\partial I+\left(\begin{matrix}-\frac{1}{1+e^{x}}&0\\\
\frac{e^{x}}{1-e^{2x}}&\frac{1}{-1+e^{x}}\end{matrix}\right).$

Letting $\pi_{K}$ and $\pi_{Q}$ be

$\displaystyle\pi_{K}=\left(\begin{matrix}e^{2x}-1&0\\\
0&1-e^{2x}\end{matrix}\right),\qquad\pi_{Q}=\left(\begin{matrix}e^{2x}-1&0\\\
0&e^{2x}-1\end{matrix}\right)$

we get

$\displaystyle\bar{K}=\pi_{K}K=\left(\begin{matrix}-e^{x}\left(-1+e^{x}\right)&0\\\
-e^{x}&e^{x}+e^{2x}\end{matrix}\right)+\left(\begin{matrix}\left(-1+e^{x}\right)\left(1+e^{x}\right)&0\\\
0&1-e^{2x}\end{matrix}\right)\partial\in\mathcal{W}$

and

$\displaystyle\bar{Q}=Q\circ\pi_{Q}=\left(\begin{matrix}1-e^{x}+2e^{2x}&0\\\
-e^{x}&1+e^{x}+2e^{2x}\end{matrix}\right)+\left(\begin{matrix}-1+e^{2x}&0\\\
0&-1+e^{2x}\end{matrix}\right)\partial\in\mathcal{W}.$

Replacing $e^{\alpha x}$ by $\top_{\alpha}^{H}=\top_{\alpha}$ and $\partial$
by $z$ we get the corresponding translation operators

$\displaystyle\flat(\bar{K})=\left(\begin{matrix}-z&0\\\
0&z\end{matrix}\right)+\top_{1}\left(\begin{matrix}1&0\\\
-1&1\end{matrix}\right)+\top_{2}\left(\begin{matrix}z-1&0\\\
0&1-z\end{matrix}\right)$

and

$\displaystyle\flat(\bar{Q})=\left(\begin{matrix}1-z&0\\\
0&1-z\end{matrix}\right)+\top_{1}\left(\begin{matrix}-1&0\\\
-1&1\end{matrix}\right)+\top_{2}\left(\begin{matrix}z+2&0\\\
0&z+2\end{matrix}\right).$

As predicted by Theorem 5.13,

$\displaystyle\Lambda=\frac{z}{p_{0}(z)}\circ\flat(\bar{Q})\circ\flat(\bar{K})\circ\frac{1}{z}=\left(\begin{matrix}\dfrac{z^{3}+5z^{2}+6z}{z(z+2)(z+3)}&0\vspace{1mm}\\\
0&\dfrac{-z^{3}-5z^{2}-6z}{z(z+2)(z+3)}\end{matrix}\right)$
$\displaystyle\hphantom{\Lambda=}{}+\top_{1}\left(\begin{matrix}0&0\vspace{1mm}\\\
\dfrac{2z^{2}+6z+4}{z(z+1)(z+2)}&0\end{matrix}\right)+\top_{2}\left(\begin{matrix}\dfrac{-2z^{3}-10z^{2}-12z}{z(z+2)(z+3)}&0\vspace{1mm}\\\
\dfrac{-2z-4}{z(z+1)(z+2)}&\dfrac{2z^{3}+10z^{2}+12z}{z(z+2)(z+3)}\end{matrix}\right)$
$\displaystyle\hphantom{\Lambda=}{}+\top_{3}\left(\begin{matrix}\dfrac{4z+12}{z(z+2)(z+3)}&0\vspace{1mm}\\\
\dfrac{-2z^{2}-4z-2}{z(z+1)(z+2)}&\dfrac{4z+12}{z(z+2)(z+3)}\\\
\end{matrix}\right)$
$\displaystyle\hphantom{\Lambda=}{}+\top_{4}\left(\begin{matrix}\dfrac{z^{3}+5z^{2}+2z-8}{z(z+2)(z+3)}&0\vspace{1mm}\\\
0&\dfrac{-z^{3}-5z^{2}-2z+8}{z(z+2)(z+3)}\end{matrix}\right)$

satisfies $\psi\Lambda=\hat{\pi}\psi$ ($\hat{\pi}=\pi_{Q}\pi_{K}$).

######  Remark 6.1.

This is a good example for demonstrating ad-nilpotency and how it has changed
in this non-commutative context. Using $L$, $\Lambda$, $p(z)$ and $\pi_{K}$,
$\pi_{Q}$ as above, it is indeed true that

$\displaystyle\textup{ad}_{\hat{\pi}}^{n}L\psi=\psi\textup{ad}_{\Lambda}^{n}p$

(with the operator in $z$ acting from the right as usual). However, contrary
to our expectation from the commutative case in which the order of the
operator on the left decreases to zero when $n$ gets large,
$\textup{ad}_{\hat{\pi}}^{n}L$ is an operator of order $2$ for every $n$. This
happens because the action of iterating $\textup{ad}_{\hat{\pi}}$ on the
leading coefficient of $L$ itself is not nilpotent. On the other hand, if
instead of $\pi_{K}$ and $\pi_{Q}$ we had chosen

$\displaystyle\pi_{K}^{*}=\pi_{Q}^{*}=\big{(}e^{2x}-1\big{)}I,\qquad\hat{\pi}^{*}=\pi_{K}^{*}\pi_{Q}^{*}=\big{(}e^{2x}-1\big{)}^{2}I$

(intentionally selected so as to commute with the coefficients of any
operator) then Theorem 5.13 would have produced a different operator
$\Lambda^{*}$ satisfying $\hat{\pi}^{*}\psi=\psi\Lambda^{*}$. In this case,
because the order is lowered at each iteration,
$\textup{ad}_{\hat{\pi}^{*}}^{3}L=0$. This is not particularly surprising or
special as the same could be said if $\hat{\pi}^{*}$ were replaced by any
function of the form $f(x)I$. However, because of the correspondence in
Theorem 5.14, we can conclude from this that
$\textup{ad}_{\Lambda^{*}}^{3}p(z)=0$ also. That is more interesting because
in general repeatedly taking the commutator of the operator $\Lambda^{*}$ with
some function will not produce the zero operator, even if all coefficients are
assumed to commute. Nevertheless, with these specific choices everything
cancels out leaving exactly zero when $\textup{ad}_{\Lambda^{*}}$ is applied
three times to the function $p(z)$ that is the eigenvalue above.

##  7 Conclusions and remarks

Given a choice of an invertible $N\times N$ matrix $H$ and $MN$-dimensional
space $W^{*}$ of vector-valued finitely supported distributions, this paper
sought to produce a bispectral triple $(L,\Lambda,\psi)$ where $L$ is a
differential operator acting from the left, $\Lambda$ is a differential-
translation operator acting from the right and $\psi$ is a common
eigenfunction that is asymptotically of the form $e^{xzH}$ satisfying the
“conditions” generated by the distributions. In the scalar case, this was
achieved in [23] for any choice of $W^{*}$. In the matrix generalization
above, however, the construction only works given Assumptions 3.6 and 5.3. The
bispectral triples produced given those two assumptions include many new
examples both in the form of the eigenfunction (asymptotically equal to
$e^{xzH}$) and the fact that the matrix-coefficient operator $\Lambda$ may
involve translation in $z$ as well as differentiation in $z$. More
importantly, this investigation yielded some observations that may be useful
in future studies of bispectrality in a non-commutative context.

This paper sought to develop a general construction of bispectral triples with
matrix-valued eigenvalues and also to understand what obstructions there might
be to generalizing the construction from [23] to the matrix case. It is
interesting to note that these seemingly separate goals both turn out to
depend on the non-commutativity of the ring $\mathcal{A}$ of functions that
stabilize the point in the Grassmannian. Since $\mathcal{A}$ is also the ring
of eigenvalues for the operators in $x$, it is not a surprise that by letting
its elements be matrix-valued gives us a non-commutative ring $\\{L_{p}\colon
p\in\mathcal{A}\\}$ of operators sharing the eigenfunction $\psi$. It was not
clear at first that the ring $\mathcal{A}$ would also be the source of the
obstruction to producing bispectral triples. However, the construction of
operators in $z$ sharing $\psi$ as eigenfunction uses the assumption that $H$
is an element of $\mathcal{A}$. (Specifically, this is used in the proof of
Lemma 5.10.) It is interesting to note that this also depends on the fact that
we considered a matrix-valued stabilizer ring.

Section 5.3 explored the extent to which the property of ad-nilpotency, which
has been a feature of papers on the bispectral problem since it was first
noted in [8], continues to apply in the case of matrix coefficient operators.
It is still the case that the operator formed by iterating the adjoint of one
of the operators in a bispectral triple on the eigenvalue of the other
operator has the same action on the eigenfunction as the operator formed by
iterating the adjoint action of its eigenvalue on the other operator. However,
unlike the scalar case, only if additional assumptions about the coefficients
of the operators are met will either of those be zero for a large enough
iterations.

It was previously observed [2, 4, 32] (see also [14, 15, 16, 17, 18]) that
requiring the operators in $x$ and $z$ to act on the eigenfunctions from
opposite side resulted in a form of bispectrality whose structure and
applications more closely resembled that in the scalar case. Because
generalizing the scalar results of [23] necessitated also requiring that the
eigenvalues and finitely-supported distributions act from the same side as the
operators acting in the same variables, this previous observation can now be
extended to those other objects as well.

This is the first time that the bispectral anti-isomorphism was used to
construct bispectral operators in a context involving operators with matrix
coefficients, see section 5.1 (cf. [12]). Some modifications were necessary
since the rather general construction in [1] assumed that there were no zero-
divisors so that formal inverses could be introduced. In addition, the use of
the method here depended on the convention of considering operators in $x$ and
$z$ to be acting from opposite sides and required that the coefficients of the
operators on which it acted were taken from the centralizer of $H$. The most
interesting difference may have been that because the operators in $z$ are
acting from the right rather than the left, the map actually preserves the
order of a product.

Unlike the scalar case, not every choice of distributions $W^{*}$ corresponds
to a bispectral triple. For instance, if Assumption 3.6 fails to be met then
there simply is no wave function $\psi$ satisfying the conditions. More
interestingly, if Assumption 3.6 is met but Assumption 5.3 is not then there
is a wave function that is an eigenfunction for a ring of differential
operators in $x$, but Theorem 5.13 does not produce a corresponding operator
in $z$. In fact, as Section 6.1 demonstrates, in at least some cases there
actually is no bispectral triple of the form considered above which includes
that wave function. This is very different than the scalar case.
Unfortunately, this paper does not entirely answer the question of which
choices of matrix $H$ and distributions $W^{*}$ produce a wave function $\psi$
that is part of a bispectral triple of the type considered here. In
particular, this paper does not show or claim that there cannot be a
differential-difference operator $\Lambda$ in $z$ having $\psi$ as an
eigenfunction with $x$-dependent eigenvalue when $H\not\in\mathcal{A}$.

Arguably, some of the non-commutativity involved in the construction above was
“artificially inserted” in the form of the choice of the matrix $H$. If one
chooses to consider only the case $H=I$, then Assumption 5.3 is automatically
met and Theorems 4.7 and 5.13 produce a bispectral triple for any $W^{*}$
satisfying Assumption 3.6. Since any operator $L_{p}$ produced by the
construction above for some choice of $H$ can be produced using a different
choice of distributions but with $H=I$, one may conclude that each of these
operators is part of a bispectral triple. (In other words, the obstruction to
bispectrality that is visible when one seeks a bispectral triple for a given
wave function $\psi$ disappears if one instead focuses on the operator $L_{p}$
and seeks a corresponding bispectral triple.) However, it seems plausible that
some examples of bispectrality to be considered in the future will involve
vacuum eigenfunctions that are necessarily non-commutative (unlike these
examples in which the non-commutativity of the vacuum eigenfunctions can
always be eliminated through a change of variables), and the observations and
results above will prove useful in those contexts.

### Acknowledgements

The author thanks the College of Charleston for the sabbatical during which
this work was completed, Maarten Bergvelt and Michael Gekhtman for
mathematical assistance as well as serving as gracious hosts, Chunxia Li for
carefully reading and commenting on early drafts, and the referees for their
advices.

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