Quantization of Calogero–Painlevé System

Quantization of Calogero–Painlevé System and
Multi-Particle Quantum Painlevé Equations II–VI

Fatane MOBASHERAMINI a and Marco BERTOLA ab F. Mobasheramini and M. Bertola

a) Department of Mathematics and Statistics, Concordia University,
a) 1455 de Maisonneuve W., Montreal, QC H3G 1M8, Canada
<EMAIL_ADDRESS><EMAIL_ADDRESS>
b) SISSA, Area of Mathematics, via Bonomea 265, Trieste, Italy

Received March 19, 2021, in final form August 31, 2021; Published online
September 07, 2021

We consider the isomonodromic formulation of the Calogero–Painlevé multi-
particle systems and proceed to their canonical quantization. We then proceed
to the quantum Hamiltonian reduction on a special representation to radial
variables, in analogy with the classical case and also with the theory of
quantum Calogero equations. This quantized version is compared to the
generalization of a result of Nagoya on integral representations of certain
solutions of the quantum Painlevé equations. We also provide multi-particle
generalizations of these integral representations.

quantization of Painlevé; Calogero–Painlevé; Harish-Chandra isomorphism

70H08; 81R12

##  1 Introduction

The six Painlevé equations were discovered by [12, 13, 24] as a result of the
search for second-order ordinary differential equations in the complex plane,
whose only movable singularities are poles, and are not generally solvable in
terms of elementary functions. These equations were subject of extensive
scrutiny by mathematicians and physicists, and they find a wide range of
applications. These equations also arise as reductions of soliton equations
[10], which are solvable by inverse scattering. Other applications include
quantum gravity and string theory [4, 5, 14], $\beta$-models [3], topological
field theories [6], random matrices [11, 26] and stochastic growth processes
[19].

Historically, the Hamiltonian structure of the Painlevé equations was studied
in [7, 20, 23] and they all can be written as a time-dependent Hamiltonian
system

$\ddot{q}=-V(q;t),$

for some potential function $V$ of dependent and independent variables.

On the other hand, these equations admit a natural generalization to multi-
particle Hamiltonian systems with an interaction of Calogero type (rational,
trigonometric, or elliptic). This approach was pioneered by K. Takasaki [25]
as an attempt at de-autonomization of a result by Inozemtsev [15]. The result
yields a new system of multi-component Hamiltonian operators that was named
Calogero–Painlevé correspondence.

The integrability of this system was conjectured by Takasaki and was shown in
[2] via an isomonodromic formulation in terms of a Lax pair of $2N\times 2N$
matrices, where $N$ is the number of particles.

On a parallel track, the integrability of the classical Calogero–Painlevé
equations has attracted increasing interest about their quantization. As a
result, the discussion about the integrability of such a system and their
corresponding solutions has appeared in the work of mathematicians and
mathematical physicists such as H. Nagoya [22], K. Okamoto [23], A. Zabrodin
and A. Zotov [27].

In [27], A. Zabrodin and A. Zotov show that all Painlevé equations I–VI can be
presented in the form of the non-stationary Schrödinger equation in imaginary
time

$\partial_{t}\Psi=H\Psi$

with $H$ the standard Schrödinger operator

$H=\frac{1}{2}\partial_{z}^{2}+V(z,t),$ (1.1)

which is a natural quantization of the classical Calogero–Painlevé Hamiltonian
operator

$H(p,q,t)=\frac{p^{2}}{2}+V(q;t).$

In this article, we follow a similar logic starting from the Hamiltonian
formulation of the multi-particle Hamiltonians in [2], with a different
approach to the system that is called the quantum Calogero–Painlevé
Hamiltonian system. This task is achieved by applying the canonical
quantization of the form (described in details in Section 2)

$\displaystyle q_{ij}\longrightarrow q_{ij}\qquad\text{and}\qquad
p_{ij}\longrightarrow\hbar\frac{\partial}{\partial q_{ji}}$

to the Hamiltonian operators in isomonodromic formulation introduced in [2].
As a result of this quantization, we obtain a multi-particle quantized
Hamiltonian system that satisfies the Schrödinger equation of the same form as
(1.1).

We also compare the result of this quantization to what H. Nagoya [22]
introduces as quantum Painlevé Hamiltonian system for a single particle wave
function with coordinate $z$. We generalize these Hamiltonian operators to the
case of $N$ particles with coordinates $z_{\rho}$, $\rho=1,\dots,N$.

We extend the integral representation of solutions given in [22] to the multi-
component integral solution of the Schrödiner equation for $N$-particle
quantum Painlevé equations. These integral representations are defined only
for the quantum Painlevé equation II–VI; these solutions should be understood
as the quantum counterpart of rational solutions and it is well known that the
first Painlevé equation does not admit rational solutions, which heuristically
explains the absence of integral representations for solutions thereof.
Intriguingly, these integral representations are presented as some type of
$\beta$ integrals that appear in other areas such as conformal field theory
[17], $\beta$-ensembles [1], the theory of orthogonal polynomials, and
hypergeometric functions.

Finally, we show that under some constraints on the parameters of the
Hamiltonian operators obtained in the first two sections of this article these
generalized Nagoya integrals provide solutions for the quantization of the
multi-particle Hamiltonian systems described in the first part.

##  2 Isomonodromic formulation and quantization

In [2] the authors provided the answer to a conjecture proposed by K. Takasaki
in [25]; Takasaki considered the de-autonomization of the Calogero systems
proposed by Inozemtsev [15] by observing that for Painlevé VI the rank-one
Inozemtsev system reduces to the autonomous version of the Hamiltonian form of
Painlevé VI that was first written by Gambier and then rediscovered by Manin.
This leads to the postulation of the deautonomized versions, and the
conjecture that they should be describing isomonodromic deformations of an
appropriate system. For this reason, Takasaki coined the term
“Painlevé–Calogero correspondence”.

The core idea of [2] is as follows; they consider a (complexified) phase space
consisting of pairs of $N\times N$ matrices $\mathbf{p}$, $\mathbf{q}$,
identified with the cotangent bundle of $X={\rm Mat}_{N\times N}(\mathbb{C})$.
The (complex) symplectic form is then

$\omega=\operatorname{Tr}(\mathrm{d}\mathbf{p}\wedge\mathrm{d}\mathbf{q})=\sum_{i,j}\mathrm{d}p_{ij}\wedge\mathrm{d}q_{ji}\quad\Leftrightarrow\quad\\{p_{ab},q_{cd}\\}=\delta_{ad}\delta_{bc}.$
(2.1)

Consider an Hamiltonian $H(\mathbf{p},\mathbf{q})$ which is conjugation
invariant, namely

$\displaystyle
H(\mathbf{p},\mathbf{q})=H\big{(}C\mathbf{p}C^{-1},C\mathbf{q}C^{-1}\big{)}\qquad\hbox{with}\quad
C\in{\rm GL}_{N}(\mathbb{C}).$

Then Noether’s theorem guarantees the conservation of the associated momenta
$M=[\mathbf{p},\mathbf{q}]$ for this group hamiltonian action. This allows us
to fix a particular value of the momentum $M$ and investigate the reduced
system on the leaf of this value. If the momentum $M$ is fixed to be of the
form

$\displaystyle M=[\mathbf{p},\mathbf{q}]={\rm i}g\big{(}\mathbf{1}-v^{\rm
T}v\big{)},\qquad\hbox{with}\quad v:=(1,\dots,1),$ (2.2)

then one can apply a theorem used in the classical theory of Calogero system
and due to Kazhdan, Kostant, and Sternberg [18]. It states that we can
diagonalize $\mathbf{q}=CXC^{-1}$ with
$X=\operatorname{diag}(x_{1},\dots,x_{n})$ in such a way that the matrix
$Y=C^{-1}\mathbf{p}C$ is of the form

$\displaystyle Y=\operatorname{diag}(y_{1},\dots,y_{N})+\left[\frac{{\rm
i}g}{x_{j}-x_{k}}\right]_{j,k=1}^{N}.$

The variables $y_{j}$’s are the momenta conjugated to the eigenvalues $x_{j}$
in the reduced system. Then the first result of [2] was that all the
Calogero–Painlevé systems of [25] are the reduction on the particular value of
the momentum (2.2) of a list of conjugation-invariant Hamiltonian systems:

$\displaystyle\widetilde{H}_{\rm
I}=\operatorname{Tr}\left(\frac{\mathbf{p}^{2}}{2}-\frac{\mathbf{q}^{3}}{2}-\frac{t\mathbf{q}}{4}\right),$
$\displaystyle\widetilde{H}_{\rm
II}=\operatorname{Tr}\left(\frac{\mathbf{p}^{2}}{2}-\frac{1}{2}\left(\mathbf{q}^{2}+\frac{t}{2}\right)^{2}-\theta\mathbf{q}\right),$
$\displaystyle t\widetilde{H}_{\rm
III}=\operatorname{Tr}\left(\mathbf{p}^{2}\mathbf{q}^{2}-\big{(}\mathbf{q}^{2}+(\theta_{0}-\theta_{1})\mathbf{q}-t\big{)}\mathbf{p}-\theta_{1}\mathbf{q}\right),$
$\displaystyle\widetilde{H}_{\rm
IV}=\operatorname{Tr}\left(\mathbf{p}\mathbf{q}(\mathbf{p}-\mathbf{q}-t)+\theta_{0}\mathbf{p}-(\theta_{0}+\theta_{1})\mathbf{q}\right),$
$\displaystyle t\widetilde{H}_{\rm
V}=\operatorname{Tr}\left(\mathbf{p}(\mathbf{p}+t)\mathbf{q}(\mathbf{q}-1)+(\theta_{0}-\theta_{2})\mathbf{p}\mathbf{q}+\theta_{2}\mathbf{p}+(\theta_{0}+\theta_{1})t\mathbf{q}\right),$
$\displaystyle t(t-1)\widetilde{H}_{\rm
VI}=\operatorname{Tr}\bigg{(}\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}-t\mathbf{p}\mathbf{q}^{2}\mathbf{p}+t\mathbf{p}\mathbf{q}\mathbf{p}-\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}-\theta\mathbf{q}\mathbf{p}\mathbf{q}+t(\theta_{0}+\theta_{1})\mathbf{p}\mathbf{q}$
$\displaystyle\hphantom{t(t-1)\widetilde{H}_{\rm
VI}=}{}+(\theta_{0}+\theta_{t})\mathbf{p}\mathbf{q}-\theta_{0}t\mathbf{p}-\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}\mathbf{q}\bigg{)},$
(2.3)

where $\theta$, $\theta_{0}$, $\theta_{1}$, $\theta_{2}$, $\theta_{t}$, and
$k$ are arbitrary parameters in $\mathbb{C}$. These Hamiltonians should be
thought of as non-commutative polynomials in $\mathbf{p}$, $\mathbf{q}$
generalizing the Okamoto Hamiltonians for the six Painlevé equations.

They showed that these Hamiltonians describe the isomonodromic deformations of
an ODE in the $z$-plane for a matrix $\Phi(z)$ of size $2N\times 2N$, which
reduces, for $N=1$, to the classical Lax pair formulation for Painlevé
equations (see, e.g., [16]).

To give a meaningful nontrivial example, we consider the Painlevé VI case:

$\begin{cases}\displaystyle\frac{\partial\Phi}{\partial
z}=\left(\frac{A_{0}}{z}+\frac{A_{1}}{z-1}+\frac{A_{t}}{z-t}\right)\Phi={\bf
A}(z)\Phi,\vspace{1mm}\\\ \displaystyle\frac{\partial\Phi}{\partial
t}=-\left(\frac{A_{t}}{z-t}+B\right)\Phi={\bf B}(z)\Phi,\end{cases}$

where the matrices are explicitly given by

$\displaystyle
A_{0}:=\left[\begin{matrix}-1-\theta_{{t}}&\displaystyle\frac{\mathbf{q}}{t}-1\\\
0&0\end{matrix}\right],\qquad
A_{1}:=\left[\begin{matrix}-\mathbf{q}\mathbf{p}+\displaystyle\frac{1}{2}(k+\theta)&1\vspace{1mm}\\\
(\theta-\mathbf{q}\mathbf{p})\mathbf{q}\mathbf{p}+\displaystyle\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}&\mathbf{q}\mathbf{p}+\displaystyle\frac{1}{2}(k-\theta)\end{matrix}\right],$
$\displaystyle
A_{t}:=\left[\begin{matrix}\mathbf{q}\mathbf{p}-\theta_{0}&-\displaystyle\frac{\mathbf{q}}{t}\\\
t(-\theta_{0}+\mathbf{p}\mathbf{q})\mathbf{p}&-\mathbf{p}\mathbf{q}\end{matrix}\right],\qquad
B:=\left[\begin{matrix}\displaystyle\frac{t([\mathbf{q},\mathbf{p}]_{+}-\theta_{0})+\theta\mathbf{q}-[\mathbf{q}\mathbf{p},\mathbf{q}]_{+}}{t(t-1)}&0\\\
-\theta_{0}\mathbf{p}+\mathbf{p}\mathbf{q}\mathbf{p}&0\end{matrix}\right],$

where $\theta=\theta_{0}+\theta_{1}+\theta_{t}$. The expression $[X,Y]_{+}$
stands here for the anti-commutator of the noncommutative symbols $X$, $Y$,
namely $[X,Y]_{+}=XY+YX$. The partitioning is in $N\times N$ block, and all
scalars are automatically considered mutliple of the identity matrix of size
$N$. The isomonodromic equations consist in the “zero-curvature” equations for
the pair

$\displaystyle\frac{\partial{\bf A}(z)}{\partial t}-\frac{\partial{\bf
B}(z)}{\partial z}+[{\bf A}(z),{\bf B}(z)]=0,$

and, with some elementary algebra, they become the following evolutionary
system for the operators $\mathbf{p}$, $\mathbf{q}$:

$\displaystyle\begin{cases}\dot{\mathbf{q}}=\mathcal{A}(\mathbf{q},\mathbf{p}),\\\
\dot{\mathbf{p}}=\mathcal{B}(\mathbf{q},\mathbf{p}),\end{cases}$ (2.4)

where the non-commutative polynomials $\mathcal{A}$, $\mathcal{B}$ are given
by

$\displaystyle
t(t-1)\mathcal{A}(\mathbf{q},\mathbf{p}):=-\theta_{0}t+(\theta_{0}+\theta_{t})\mathbf{q}+(\theta_{0}+\theta_{1})t\mathbf{q}-\theta\mathbf{q}^{2}-2\mathbf{q}\mathbf{p}\mathbf{q}$
$\displaystyle\hphantom{t(t-1)\mathcal{A}(\mathbf{q},\mathbf{p}):=}{}+t[\mathbf{p},\mathbf{q}]_{+}-\big{[}t\mathbf{p},\mathbf{q}^{2}\big{]}_{+}+[\mathbf{q}\mathbf{p}\mathbf{q},\mathbf{q}]_{+},$
$\displaystyle
t(t-1)\mathcal{B}(\mathbf{q},\mathbf{p}):=\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}-(\theta_{0}+\theta_{t})\mathbf{p}-(\theta_{0}+\theta_{1})t\mathbf{p}+\theta[\mathbf{q},\mathbf{p}]_{+}-t\mathbf{p}^{2}$
$\displaystyle\hphantom{t(t-1)\mathcal{B}(\mathbf{q},\mathbf{p}):=}{}+t\big{[}\mathbf{q},\mathbf{p}^{2}\big{]}_{+}+\mathbf{p}\big{(}2\mathbf{q}-\mathbf{q}^{2}\big{)}\mathbf{p}-[\mathbf{q},\mathbf{p}\mathbf{q}\mathbf{p}]_{+}.$
(2.5)

The key observation, which is the initial thrust of our present paper, is the
following: in the above derivation of the zero curvature equations the symbols
$\mathbf{p}$, $\mathbf{q}$ can be taken in some arbitrary non-commutative
algebra. In the case of [2] where $\mathbf{p}$, $\mathbf{q}$ are matrices, the
above equations turn out, by inspection, to be Hamiltonian equations with
respect to the symplectic structure (2.1) with an Hamiltonian given by

$\displaystyle
t(t-1)H=\operatorname{Tr}\bigg{(}\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}-t\mathbf{p}\mathbf{q}^{2}\mathbf{p}+t\mathbf{p}\mathbf{q}\mathbf{p}-\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}-\theta\mathbf{q}\mathbf{p}\mathbf{q}+t(\theta_{0}+\theta_{1})\mathbf{p}\mathbf{q}$
$\displaystyle\hphantom{t(t-1)H=}{}+(\theta_{0}+\theta_{t})\mathbf{p}\mathbf{q}-\theta_{0}t\mathbf{p}-\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}\mathbf{q}\bigg{)}.$

Analogous considerations apply to each of the other cases, with the
Hamiltonians given in (2.3).

Quantization. In view of the considerations above, we want to consider the
canonical quantization of the symplectic structure (2.1). The main logic is
that we keep equations (2.4) (and the other equations corresponding to each of
the other cases listed in [2]) and seek a Hamiltonian formulation with quantum
Hamiltonians. The canonical quantization in “Schrödinger” representation
amounts to considering the entries of $\mathbf{q}$ as multiplication operators
and the entries of $\mathbf{p}$ corresponding differential operators as
follows

$q_{ij}\longrightarrow q_{ij}\qquad\text{and}\qquad
p_{ij}\longrightarrow\hbar\frac{\partial}{\partial q_{ji}}.$ (2.6)

The effect of this canonical quantization is that we cannot simply take the
expressions (2.3) as Hamiltonians generating the relevant equations of motions
like (2.4) because there are issues of normal ordering. To explain the issue
we point out that in the classical case the expressions
$\operatorname{Tr}(\mathbf{p}\mathbf{q})$ and
$\operatorname{Tr}(\mathbf{q}\mathbf{p})$ coincide, but if $\mathbf{p}$,
$\mathbf{q}$ are quantum operators (2.6) then these two expressions differ.
This should explain why the quantum version of the Hamiltonians (2.3) will be
slightly different due to the fact that the correct scheme depends on the non-
commutativity of the traces in this case. Note that the commutation relations
that lead to these Hamiltonian operators read the following equations for each
of the Calogero–Painlevé equations:

$\displaystyle\hbar\dot{\mathbf{q}}=[H_{J},\mathbf{q}]\qquad\text{and}\qquad\hbar\dot{\mathbf{p}}=[H_{J},\mathbf{p}],\qquad
J\in\\{{\rm I},\dots,{\rm VI}\\}.$

The correct Hamiltonian operators to which we apply the quantization, are the
following operators

$\displaystyle t\widetilde{H}_{\rm
III}=\operatorname{Tr}\bigg{(}\frac{\mathbf{p}^{2}\mathbf{q}^{2}+\mathbf{q}^{2}\mathbf{p}^{2}}{2}-\frac{\mathbf{q}^{2}\mathbf{p}+\mathbf{p}\mathbf{q}^{2}}{2}-(\theta_{0}-\theta_{1})\mathbf{q}\mathbf{p}+t\mathbf{p}-\theta_{1}\mathbf{q}\bigg{)},$
$\displaystyle\widetilde{H}_{\rm
IV}=\operatorname{Tr}\bigg{(}\mathbf{p}\mathbf{q}\mathbf{p}-\frac{\mathbf{p}\mathbf{q}^{2}+\mathbf{q}^{2}\mathbf{p}}{2}-t\mathbf{p}\mathbf{q}+\theta_{0}p-(\theta_{0}+\theta_{1})\mathbf{q}\bigg{)},$
$\displaystyle t\widetilde{H}_{\rm
V}=\operatorname{Tr}\bigg{(}\frac{\mathbf{p}^{2}\mathbf{q}^{2}+\mathbf{q}^{2}\mathbf{p}^{2}}{2}-\frac{\mathbf{p}^{2}\mathbf{q}+\mathbf{q}\mathbf{p}^{2}}{2}+\frac{t\big{(}\mathbf{p}\mathbf{q}^{2}+\mathbf{q}^{2}\mathbf{p}\big{)}}{2}$
$\displaystyle\hphantom{t\widetilde{H}_{\rm
V}=}{}+(\theta_{0}-\theta_{2}-t)\mathbf{p}\mathbf{q}+\theta_{2}\mathbf{p}+(\theta_{0}+\theta_{1})t\mathbf{q}\bigg{)},$
$\displaystyle t(t-1)\widetilde{H}_{\rm
VI}=\operatorname{Tr}\bigg{(}\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}-t\mathbf{p}\mathbf{q}^{2}\mathbf{p}+t\mathbf{p}\mathbf{q}\mathbf{p}-\frac{\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}+\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p}}{2}-\theta\mathbf{q}\mathbf{p}\mathbf{q}$
$\displaystyle\hphantom{t(t-1)\widetilde{H}_{\rm
VI}=}{}+t(\theta_{0}+\theta_{1})\mathbf{p}\mathbf{q}+(\theta_{0}+\theta_{t})\mathbf{p}\mathbf{q}-\theta_{0}t\mathbf{p}-\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}\mathbf{q}\bigg{)},$
(2.7)

whereas the Painlevé I and Painlevé II Hamiltonians remain formally the same.
Note that we use $\mathbf{p}$, $\mathbf{q}$ here and below to denote the
quantum operators, without further notice.

The operators above are determined, by inspection, requiring that the
evolution equations for $\mathbf{p}$, $\mathbf{q}$ remain of the same form as
those in [2].

######  Example 2.1.

We compute
$[\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})]$
both quantistically and classically to show the origin of the difference in
the quantum Hamiltonians (2.7). We start with the classical computation where
$\\{p_{ij},q_{k\ell}\\}=\delta_{i\ell}\delta_{jk}$:

$\big{\\{}\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})\big{\\}}=2\mathbf{p}\mathbf{q}\mathbf{p}.$

In computing this we have used also the cyclicity of the trace.

Viceversa, considering the quantized operators, the same expression yields

$\displaystyle[\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})]_{ij}=\sum_{\alpha,\beta,\mu,\nu}\left[p_{ij},p_{\alpha\beta}q_{\beta\mu}p_{\mu\nu}q_{\nu\alpha}\right]$
$\displaystyle\hphantom{[\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})]_{ij}}{}=\sum_{\alpha,\beta,\mu,\nu}p_{\alpha\beta}\left(p_{ij}q_{\beta\mu}p_{\mu\nu}q_{\nu\alpha}-q_{\beta\mu}p_{\mu\nu}q_{\nu\alpha}p_{ij}\right)$

we add and subtract $q_{\beta\mu}p_{ij}p_{\mu\nu}q_{\nu\alpha}$ to the
expression inside the bracket, combining the terms and using the commutation
relation $[p_{ij},q_{kl}]=\hbar\delta_{il}\delta_{jk}$:

$\displaystyle[\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})]_{ij}=\hbar\sum_{\alpha,\beta,\mu,\nu}p_{\alpha\beta}\left(\delta_{i\mu}\delta_{j\beta}p_{\mu\nu}q_{\nu\alpha}+\delta_{i\alpha}\delta_{j\nu}q_{\beta\mu}p_{\mu\nu}\right)$
$\displaystyle\hphantom{[\mathbf{p},\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})]_{ij}}{}=\big{(}\hbar\mathbf{p}^{2}\mathbf{q}+\hbar\mathbf{p}\mathbf{q}\mathbf{p}\big{)}_{ij}=\big{(}2\hbar\mathbf{p}\mathbf{q}\mathbf{p}+\hbar^{2}\mathbf{p}\big{)}_{ij}.$

Since the desired term in the equation of motion is
$2\hbar\mathbf{p}\mathbf{q}\mathbf{p}$ we need to replace the term
$\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q})$ in the
Hamiltonian with a “symmetrized” version
$\frac{1}{2}\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}+\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p})$.
Indeed, one then similarly computes:

$\displaystyle\left[\mathbf{p},\frac{1}{2}\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p}\mathbf{q}+\mathbf{q}\mathbf{p}\mathbf{q}\mathbf{p})\right]=2\hbar\mathbf{p}\mathbf{q}\mathbf{p},$

which is one of the terms that appears in the expression of $\mathcal{A}$ in
equation of motion (2.5).

###  2.1 Quantization in radial form

Having established the correct context of the quantization of the
isomonodromic equations we proceed now to the quantum version of the
Kazhdan–Kostant–Sternberg reduction (2.2). The first step is to express the
quantum Hamiltonians (2.7) in terms of the eigenvalues. The fact that this is
at all possible is simply a consequence of the invariance of the Hamiltonians
under conjugations. To do so, we need to use the Harish-Chandra homomorphism
[8].

######  Definition 2.2.

Let $\mathcal{M}$ be the manifold of diagonalizable matrices and denote

$\mathcal{D}(\mathcal{M})^{\rm G}\qquad\text{and}\qquad\mathcal{D}({\rm
Diag}({\rm GL}_{n}))^{\rm W}$

the adjoint-invariant subalgebra of the algebra of differential operators over
$\mathcal{M}$ and the Weyl-invariant subalgebra of the algebra of differential
operators over diagonal matrices, respectively.

The canonical isomorphism

$\mathcal{H}_{\rm c}\colon\ \mathcal{D}(M)^{\rm
G}\longrightarrow\mathcal{D}({\rm Diag}({\rm GL}_{n}))^{\rm W}$ (2.8)

is called the Harish-Chandra map.

This means the following: for a character function $\Psi(Q)$ (i.e.,
$\Psi(Q)=\Psi\big{(}GQG^{-1}\big{)}$, $G\in{\rm GL}_{n}$) and $\mathcal{L}$ a
differential operator invariant under the adjoint map we have

$(\mathcal{L}\Psi)|_{\rm Diag}=\mathcal{H}_{\rm c}(\mathcal{L})(\Psi|_{\rm
Diag}),$

where $\mathcal{H}_{\rm c}(\mathcal{L})$ is a differential operator on the
eigenvalues.

Our goal now is to make this isomorphism completely explicit and subsequently
express all Hamiltonians in (2.7) as differential operators acting on the
eigenvalues when applied to character functions or pseudo-character functions,
namely $\Psi(\mathbf{q})=\Psi\big{(}G\mathbf{q}G^{-1}\big{)}{\rm
e}^{\theta(G,\mathbf{q})}$, with $\theta$ an appropriate cocycle.

Explicit construction of Harish-Chandra isomorphism. To make the Harish-
Chandra homomorphism (2.8) explicit, we write the matrix $Q=Z+M$ with $Z$
diagonal and $M$ off-diagonal: we then act with an infinitesimal conjugation
up to order two in $M$ to diagonalize it. Concretely this means the following;
we conjugate the matrix $Q=Z+M$ by a matrix of the form $G:={\rm
e}^{A^{(1)}+A^{(2)}}$ where $A^{(1)}$ is assumed to be of first order in the
entries of $M$ and $A^{(2)}$ of second order and both are off-diagonal
matrices.

We then impose that the conjugation of $Q$ by $G$ is diagonal up to order $2$.
Then

$\displaystyle{\rm e}^{{\rm
ad}_{A^{(1)}+A^{(2)}}}(Z+M)=Z+M+\big{[}A^{(1)}+A^{(2)},Z+M\big{]}+\frac{1}{2}\big{[}A^{(1)},\big{[}A^{(1)},Z\big{]}\big{]}+\mathcal{O}(3)$
$\displaystyle\qquad{}=Z+M+\big{[}A^{(1)},Z\big{]}+\big{[}A^{(1)},M\big{]}+\big{[}A^{(2)},Z\big{]}+\frac{1}{2}\big{[}A^{(1)},\big{[}A^{(1)},Z\big{]}\big{]}+\mathcal{O}(3),$
(2.9)

where $\mathcal{O}(3)$ denotes terms of order $3$ or higher in the entries of
$M$. We need to impose that the result is a diagonal matrix up to the
indicated order. Separating the equations according to their order in $M$ we
obtain

$\displaystyle\big{[}Z,A^{(1)}\big{]}=M\qquad\text{at order 1},$ (2.10)
$\displaystyle\big{[}Z,A^{(2)}\big{]}=\big{[}A^{(1)},M\big{]}+\frac{1}{2}\big{[}A^{(1)},\big{[}A^{(1)},Z\big{]}\big{]}\qquad\text{at
order 2}.$ (2.11)

The matrices $A^{(1,2)}$ are off-diagonal, and the ${\rm ad}_{Z}$ operator is
invertible on the subspace of off-diagonal matrices so that we can solve the
two equations above to obtain

$\displaystyle A^{(1)}_{ab}=\frac{M_{ab}}{z_{a}-z_{b}},$ $\displaystyle
A^{(2)}_{ab}=-\frac{1}{2}\frac{\big{[}A^{(1)},\big{[}A^{(1)},Z\big{]}\big{]}_{ab}}{z_{a}-z_{b}}=\frac{1}{2}\frac{\big{[}A^{(1)},M\big{]}_{ab}}{z_{a}-z_{b}}=\frac{M_{ac}M_{cb}}{(z_{a}-z_{c})(z_{a}-z_{b})}.$
(2.12)

Substituting (2.12) into (2.9) we obtain a diagonal matrix $\widetilde{Z}$
which is a shift of the matrix $Z$ as follows

$\displaystyle\widetilde{Z}=Z+\frac{1}{2}\big{[}A^{(1)},M\big{]}_{D}+\mathcal{O}(3)$
$\displaystyle\hphantom{\widetilde{Z}}{}=A^{(2)}+\frac{1}{2}\operatorname{diag}\left(\sum_{c}\frac{M_{\bullet
c}}{z_{\bullet}-z_{c}}M_{c\bullet}-M_{\bullet
c}\frac{M_{c\bullet}}{z_{c}-z_{\bullet}}\right)_{\bullet=1}^{N}+\mathcal{O}(3)$
$\displaystyle\hphantom{\widetilde{Z}}{}=Z+\operatorname{diag}\left(\sum_{d}\frac{M_{\bullet
d}M_{d\bullet}}{z_{\bullet}-z_{d}}\right)_{\bullet=1}^{N}.$ (2.13)

We now show how to use the above diagonalization to second order (2.13) to
construct the Harish-Chandra homomorphism; we anticipate that the reason why
we expand up to order $2$ is that all the operators we consider are at most
quadratic in the momenta $\mathbf{p}$ and hence translate to differential
operators of order $2$. If we had to consider the Harish-Chandra homomorphism
for operators of the higher order, we would have to perform the above
diagonalization up to the corresponding order.

Space of radial functions. When considering the quantum version of the
Kazhdan–Kostant–Sternberg reduction, the choice of the special value of the
momentum $M$ (2.2) is replaced by the requirement that the quantum operators
act on specific representations. We start with a general discussion on
equivariant functions.

Let $\mathbb{V}$ be a vector space carrying a representation $\gamma$ of
$G={\rm GL}_{N}$ and let $\Psi\colon{\rm Mat}_{N\times
N}(\mathbb{C})=\operatorname{Lie}(G)\to\mathbb{V}$ an $\gamma$-equivariant
function in the sense that

$\displaystyle\Psi(Q)=\gamma\big{(}g^{-1}\big{)}\Psi\big{(}gQg^{-1}\big{)},$

where $g\in{\rm GL}_{N}(\mathbb{C})$ and $\gamma\colon{\rm
GL}_{N}(\mathbb{C})\to\operatorname{Aut}(\mathbb{V})$ is a representation. Let
us denote

$\displaystyle H_{\gamma}:=\big{\\{}\Psi\colon{\rm Mat}_{N\times
N}(\mathbb{C})\to V,\,\text{$\gamma$-equivariant}\big{\\}}.$

For simplicity, we denote by the same symbol $\gamma$ the representation of
${\rm SL}_{n}$, the corresponding representation of the Lie algebra
$\mathfrak{g}=\mathfrak{sl}_{N}$ as well as its natural extension to the
universal enveloping algebra $U(\mathfrak{g})$. We recall that the zero-weight
subspace of the ${\rm SL}_{N}$-representation $\mathbb{V}$ is

$\displaystyle\mathbb{V}(0):=\bigcap_{H\in\mathfrak{h}}\operatorname{Ker}\gamma(H),$

where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{sl}_{N}$
(traceless diagonal matrices).

######  Lemma 2.3.

Let $\mathcal{D}\subset{\rm Mat}_{N\times N}(\mathbb{C})$ consist of the
subspace of diagonal matrices. Then any equivariant function $\Psi$ restricts
to a function from $\mathcal{D}$ to $\mathbb{V}(0)$.

###### Proof.

Consider a matrix $g={\rm e}^{\epsilon H}$ with $H\in\mathfrak{h}$; then
$\Psi(Q)=\gamma\big{(}{\rm e}^{\epsilon H}\big{)}\Psi\big{(}{\rm e}^{\epsilon
H}Q{\rm e}^{-\epsilon H}\big{)}$. Restricting $Q=Z\in\mathcal{D}$ we have
$\Psi(Z)=\gamma\big{(}{\rm e}^{\epsilon H}\big{)}\Psi(Z)$. We now take the
derivative with respect to $\epsilon$ at $\epsilon=0$ and we obtain
$\gamma(H)\Psi(Z)=0$. Since $H$ is arbitrary in $\mathfrak{h}$ it follows that
$\Psi(Z)\in\mathbb{V}(0)$. ∎

Following [8] the quantum Hamiltonian reduction that corresponds to the
Kazhdan–Kostant–Sternberg orbit consists in taking a particular representation
$\gamma$ of $\mathfrak{sl}_{N}$; the main feature of the $\mathfrak{g}$-module
(which we denote by $\mathbb{V}_{\kappa}$) is that the zero weight space
$\mathbb{V}_{\kappa}(0)$ is unidimensional. We denote with ${\mathbf{c}}$ a
spanning element. Specifically, $\mathbb{V}_{\kappa}$ consists of the space of
functions of the form

$F(\xi_{1},\dots,\xi_{N})=\left(\prod_{j=1}^{N}\xi_{j}\right)^{\kappa}f\big{(}\vec{\xi}\,\big{)},$

where $f\big{(}\vec{\xi}\,\big{)}$ is a rational function with zero degree of
homogeneity. The representation of the Lie algebra $\mathfrak{sl}_{N}$ is then
the one obtained by restriction of the following $\mathfrak{gl}_{N}$
representation

$\displaystyle\gamma(\mathbb{E}_{ab})=\xi_{a}\frac{\partial}{\partial\xi_{b}},\qquad
a,b=1,\dots,N.$ (2.14)

It is easy then to see that
$\mathbb{V}_{\kappa}(0)=\mathbb{C}\big{\\{}\prod_{j=1}^{N}\xi_{j}^{\kappa}\big{\\}}$.

Keeping this in mind we illustrate the type of computations needed to compute
the extended Harish-Chandra homomorphism in the following example.

######  Example 2.4.

To illustrate the type of computations necessary, we consider the quantum
radial reduction of the operator
$\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}$. Using the form
of the quantum operators $\mathbf{p}$, $\mathbf{q}$ we obtain

$\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=\bigg{(}\sum_{\rho,\sigma,\tau}\big{(}q^{k}\big{)}_{\rho\sigma}p_{\sigma\tau}p_{\tau\rho}\bigg{)}\Psi(Q)=\bigg{(}\hbar^{2}\sum_{\rho,\sigma,\tau}\big{(}q^{k}\big{)}_{\rho\sigma}\partial_{q_{\tau\sigma}}\partial_{q_{\rho\tau}}\bigg{)}\Psi(Q).$

Since the function $\Psi$ is $\gamma$-equivariant and the operator is
${\mathrm{Ad}}$-invariant we can write $\Psi(Q)=\gamma(g)\Psi(Z)$ where $g$ is
the matrix diagonalizing $Q$, and $Z$ is the diagonal matrix of its
eigenvalues (this can be done on the set of diagonalizable matrices $Q$ whose
complement of non-diagonalizable matrices is of zero measure and hence
inessential to our considerations). We then consider matrices of the form
$Q=Z+M$ with $Z$ diagonal and $M$ off-diagonal and its diagonalization up to
order $2$ as in (2.9). We then need to perform the derivatives and, at the end
of the computation, restrict to the locus of diagonal matrices $Q=Z$. Using
equation (2.13) and the matrices $A^{(1,2)}$ introduced in (2.10) and (2.11)
we can continue the above computation by noticing that the terms involving the
multiplication operator $\mathbf{q}$ can be directly evaluated at $Q=Z$
setting $M=0$:

$\displaystyle\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=\left(\hbar^{2}\sum_{\rho,\sigma,\tau}\delta_{\rho\sigma}{z}^{k}_{\sigma}\partial_{q_{\tau\sigma}}\partial_{q_{\rho\tau}}\right)\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\tilde{Z}\big{)}=$
$\displaystyle\hphantom{\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)}{}=\left(\hbar^{2}\sum_{\sigma,\tau}z^{k}_{\sigma}\partial_{q_{\tau\sigma}}\partial_{q_{\sigma\tau}}\right)\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\tilde{Z}\big{)}.$ (2.15)

The second-order operator
$\sum_{\rho,\sigma}z^{k}_{\sigma}\partial_{q_{\sigma\rho}}\partial_{q_{\rho\sigma}}$
written in terms of $Z$, $M$ becomes the operator
$\sum_{\rho}z_{\rho}^{k}\partial_{z_{\rho}}^{2}+\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}z^{k}_{\rho}\partial_{M_{\sigma\rho}}\partial_{M_{\rho\sigma}}$;
the part involving the derivatives with respect to $z_{\rho}$ can be directly
evaluated at $Q=Z$ while we postpone the evaluation of the part involving the
derivatives in $M_{\rho,\sigma}$:

$\displaystyle\eqref{223}=\hbar^{2}\sum_{\sigma}z^{k}_{\sigma}\partial_{z_{\sigma}}^{2}\Psi(Z)+\hbar^{2}\underbrace{\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}z^{k}_{\sigma}\partial_{M_{\tau\sigma}}\partial_{M_{\sigma\tau}}\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\tilde{Z}\big{)}}_{*},$ (2.16)

where $\widetilde{Z}=Z+\operatorname{diag}\Big{(}\sum_{d}\frac{M_{\bullet
d}M_{d\bullet}}{z_{\bullet}-z_{d}}\Big{)}_{\bullet=1}^{N}$ as in (2.13).
Consider now the term marked with an asterisk: since $\widetilde{Z}-Z$ is a
quadratic expression in the entries of $M$, if we differentiate once $\Psi$ or
$\gamma$ by $M_{\rho\sigma}$, by the chain rule there will be a multiplication
by entire of $M$ in the result. Thus, subsequent evaluation at $M=0$ will
eliminate these terms. Therefore we need to consider the second-order operator
acting on $\gamma$ or $\Psi$ separately. When acting on $\Psi$ we have

$\displaystyle\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}z^{k}_{\sigma}\partial_{M_{\sigma\rho}}\partial_{M_{\rho\sigma}}\Psi\big{(}\widetilde{Z}\big{)}\bigg{|}_{M=0}=\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}z^{k}_{\sigma}\frac{\partial}{\partial{M_{\sigma\rho}}}\left(\frac{M_{\sigma\rho}\partial_{z_{\sigma}}\Psi}{z_{\sigma}-z_{\rho}}+\frac{M_{\sigma\rho}\partial_{z_{\rho}}\Psi}{z_{\rho}-z_{\sigma}}\right)\bigg{|}_{M=0}$
$\displaystyle\hphantom{\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}z^{k}_{\sigma}\partial_{M_{\sigma\rho}}\partial_{M_{\rho\sigma}}\Psi\big{(}\widetilde{Z}\big{)}\bigg{|}_{M=0}}{}=\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}z^{k}_{\sigma}\left(\frac{\partial_{z_{\sigma}}-\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}\right)\Psi(Z).$

For the computation of the second term involving the representation $\gamma$,
we note that

$\displaystyle\partial_{M_{\sigma\tau}}A^{(1)}=\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}\frac{E_{\sigma\tau}}{z_{\sigma}-z_{\tau}},\quad\partial_{M_{\tau\sigma}}\partial_{M_{\sigma\tau}}A^{(1)}=0,$
$\displaystyle\partial_{M_{\sigma\tau}}A^{(2)}=\sum_{\begin{subarray}{c}\sigma,\tau,\nu\\\
\nu\neq\sigma\neq\tau\end{subarray}}\frac{M_{\tau\nu}E_{\sigma\nu}}{(z_{\sigma}-z_{\tau})(z_{\sigma}-z_{\nu})}+\sum_{\begin{subarray}{c}\sigma,\tau,\mu\\\
\mu\neq\sigma\neq\tau\end{subarray}}\frac{M_{\mu\sigma}E_{\mu\tau}}{(z_{\mu}-z_{\sigma})(z_{\mu}-z_{\tau})},\qquad\partial_{M_{\tau\sigma}}\partial_{M_{\sigma\tau}}A^{(2)}=0.$

Therefore the action of these differential operators on the group element
${\rm e}^{-A^{(1)}-A^{(2)}}$ gives (retaining the terms up to order $2$ in $M$
in the expansion of the exponential, since all higher order terms will give
zero contribution when evaluated at $M=0$)

$\displaystyle\partial_{M_{\sigma\tau}}{\rm
e}^{-A^{(1)}-A^{(2)}}=-\partial_{M_{\sigma\tau}}A^{(1)}-\partial_{M_{\sigma\tau}}A^{(2)}+\frac{\big{[}\partial_{M_{\sigma\tau}}A^{(1)},A^{(1)}\big{]}_{+}}{2},$
$\displaystyle\partial_{M_{\tau\sigma}}\partial_{M_{\sigma\tau}}{\rm
e}^{-A^{(1)}-A^{(2)}}\bigg{|}_{M=0}=-\frac{[\mathbb{E}_{\tau\sigma},\mathbb{E}_{\sigma\tau}]_{+}}{2(z_{\sigma}-z_{\tau})^{2}}.$

Here $\mathbb{E}_{\tau\sigma}$ denote the elementary matrices. Note also that
$\mathbb{E}_{\tau\sigma}\mathbb{E}_{\sigma\tau}=\mathbb{E}_{\tau\tau}$ are
diagonal matrices and
$[\mathbb{E}_{\tau\sigma},\mathbb{E}_{\sigma\tau}]_{+}=\mathbb{E}_{\tau\tau}+\mathbb{E}_{\sigma\sigma}$.
Hence, the equation (2.16) yields

$\displaystyle\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=\hbar^{2}\sum_{\sigma}z_{\sigma}^{k}\partial_{z_{\sigma}}^{2}\Psi(Z)$
$\displaystyle\qquad{}+\hbar^{2}\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}z_{\sigma}^{k}\left(-\frac{\gamma([\mathbb{E}_{\tau\sigma},\mathbb{E}_{\sigma\tau}]_{+})}{2(z_{\sigma}-z_{\tau})^{2}}\Psi\big{(}\tilde{Z}\big{)}+\left(\frac{\partial_{z_{\sigma}}-\partial_{z_{\tau}}}{z_{\sigma}-z_{\tau}}\right)\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\tilde{Z}\big{)}\right)\bigg{|}_{M=0}.$

To complete the computation, we recall that under the assumption for the
representation space $\mathbb{V}_{\kappa}$ (see (2.14)) we easily see that
$\gamma(\mathbb{E}_{\sigma\tau}\mathbb{E}_{\tau\sigma})=\kappa(\kappa+1){\rm
Id}_{\mathbb{V}_{\kappa}(0)}$. Recall also (Lemma 2.3) that $\Psi$ evaluated
on diagonal matrices takes values in the zero weight space
$\mathbb{V}_{\kappa}(0)$. Therefore we conclude that
$\gamma([\mathbb{E}_{\tau\sigma},\mathbb{E}_{\sigma\tau}]_{+})$ reduces simply
to the multiplication by $2\kappa(\kappa+1)$.

This leads finally to the result

$\displaystyle\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=\bigg{(}\hbar^{2}\\!\sum_{\sigma}z^{k}_{\sigma}\partial_{z_{\sigma}}^{2}-\hbar^{2}\kappa(\kappa+1)\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}\frac{z^{k}_{\sigma}}{(z_{\sigma}-z_{\tau})^{2}}+\hbar^{2}\\!\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}\frac{z_{\sigma}^{k}(\partial_{z_{\sigma}}-\partial_{z_{\tau}})}{z_{\sigma}-z_{\tau}}\bigg{)}\Psi(Z).$

To make the last term more symmetric we can add and subtract
$z_{\tau}^{k}\partial_{z_{\tau}}$ and obtain

$\displaystyle\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=\bigg{(}\hbar^{2}\\!\sum_{\sigma}z^{k}_{\sigma}\partial_{z_{\sigma}}^{2}-\hbar^{2}\kappa(\kappa+1)\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}\frac{z^{k}_{\sigma}}{(z_{\sigma}-z_{\tau})^{2}}+\hbar^{2}\\!\sum_{\begin{subarray}{c}\sigma,\tau\\\
\sigma\neq\tau\end{subarray}}\frac{z_{\sigma}^{k}\partial_{z_{\sigma}}-z_{\tau}^{k}\partial_{z_{\tau}}}{z_{\sigma}-z_{\tau}}\bigg{)}\Psi(Z)$
$\displaystyle\hphantom{\operatorname{Tr}\big{(}\mathbf{q}^{k}\mathbf{p}^{2}\big{)}\Psi(Q)=}{}-\hbar^{2}\sum_{j=0}^{k-1}\sum_{\sigma}z_{\sigma}^{j}\sum_{\tau}z_{\tau}^{k-j-1}\partial_{z_{\tau}}\Psi(Z)+\hbar^{2}k\sum_{\tau}z_{\tau}^{k-1}\partial_{z_{\tau}}\Psi(Z).$

Throughout the following section, we use this method to obtain the quantized
Calogero–Painlevé II–VI Hamiltonian operators.

####  2.1.1 Quantization of Calogero–Painlevé Hamiltonians I–VI

We apply the quantum Hamiltonian KKS reduction explained in the previous
sections to the list of Hamiltonians (2.7). Moreover note that unlike the
classic case, in the quantum case we have non-commutative operator-valued
matrices $\mathbf{p}$ and $\mathbf{q}$ so that
$\operatorname{Tr}(\mathbf{p}\mathbf{q})$ is not equal to
$\operatorname{Tr}(\mathbf{q}\mathbf{p})$. For example, we have the following

$\displaystyle\operatorname{Tr}(\mathbf{p}\mathbf{q})=\operatorname{Tr}(\mathbf{q}\mathbf{p})+\hbar
N^{2},$
$\displaystyle\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p})=\operatorname{Tr}\big{(}\mathbf{q}\mathbf{p}^{2}\big{)}+\hbar
N\operatorname{Tr}(\mathbf{p}),$
$\displaystyle\operatorname{Tr}(\mathbf{q}\mathbf{p}\mathbf{q})=\operatorname{Tr}\big{(}\mathbf{q}^{2}\mathbf{p}\big{)}+\hbar
N\operatorname{Tr}(\mathbf{q}),$
$\displaystyle\operatorname{Tr}\big{(}\mathbf{p}\mathbf{q}^{2}\big{)}=\operatorname{Tr}\big{(}\mathbf{q}^{2}\mathbf{p}\big{)}+2\hbar
N\operatorname{Tr}(\mathbf{q}),$
$\displaystyle\operatorname{Tr}\big{(}\mathbf{p}^{2}\mathbf{q}^{2}\big{)}=\operatorname{Tr}\big{(}\mathbf{q}^{2}\mathbf{p}^{2}\big{)}+2\hbar
N\operatorname{Tr}(\mathbf{q}\mathbf{p})+2\hbar\operatorname{Tr}(\mathbf{q})\operatorname{Tr}(\mathbf{p})+\hbar^{2}N\big{(}1+N^{2}\big{)}.$

We start from the Hamiltonian operator corresponding to Calogero–Painlevé II
(more complicated case than Calogero–Painlevé I) and we apply it to the
$\gamma$-equivariant wave function $\Psi(Q)$, and then we apply the
quantization (2.6) to the result:

$\displaystyle\tilde{H}_{\rm
II}\Psi(Q)=\operatorname{Tr}\left(\frac{\mathbf{p}^{2}}{2}-\frac{1}{2}\left(\mathbf{q}^{2}+\frac{t}{2}\right)^{2}-\theta\mathbf{q}\right)\Psi(Q)$
$\displaystyle\hphantom{\tilde{H}_{\rm
II}\Psi(Q)}{}=\left(\frac{\hbar^{2}}{2}\sum_{\rho,\sigma}\partial_{q_{\sigma\rho}}\partial_{q_{\rho\sigma}}-\frac{1}{2}\sum_{\rho,\sigma}\left(\big{(}\delta_{\sigma\rho}^{2}q_{\rho\sigma}^{2}\big{)}+\frac{t}{2}\right)^{2}-\theta\delta_{\sigma\rho}q_{\rho\sigma}\right)\Psi(Q).$
(2.17)

Following the same logic used in the illustrative example of the operator
$\operatorname{Tr}(\mathbf{p}\mathbf{q}\mathbf{p})$ we can continue the
computation

$\displaystyle\eqref{227}=\frac{\hbar^{2}}{2}\\!\sum_{\rho,\sigma}\partial_{q_{\sigma\rho}}\partial_{q_{\rho\sigma}}\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\widetilde{Z}\big{)}-\frac{1}{2}\sum_{\rho}\\!\left(z_{\rho}^{2}+\frac{t}{2}\right)^{2}\\!\Psi(Z)-\theta\sum_{\rho}z_{\rho}\Psi(Z),\\!\\!\\!$
(2.18)

where $\widetilde{Z}=Z+\operatorname{diag}\Big{(}\sum_{d}\frac{M_{\bullet
d}M_{d\bullet}}{z_{\bullet}-z_{d}}\Big{)}_{\bullet=1}^{N}$ as in (2.13). Note
that the terms involving only the multiplication operator $\mathbf{q}$ can be
directly evaluated at $Q=Z$ setting $M=0$. The second-order operator
$\sum_{\rho,\sigma}\partial_{q_{\sigma\rho}}\partial_{q_{\rho\sigma}}$ in
terms of $Z$, $M$ becomes the operator
$\sum_{\rho}\partial_{z_{\rho}}^{2}+\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\partial_{M_{\sigma\rho}}\partial_{M_{\rho\sigma}}$;
once again, the part involving the derivatives with respect to $z_{\rho}$ can
be directly evaluated at $Q=Z$ while we postpone the evaluation of the part
involving the derivatives in $M_{\rho,\sigma}$:

$\displaystyle\eqref{228}=\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}\Psi(Z)+\frac{\hbar^{2}}{2}\underbrace{\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\partial_{M_{\sigma\rho}}\partial_{M_{\rho\sigma}}\big{(}\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\Psi\big{(}\widetilde{Z}\big{)}\big{)}}_{*}$
$\displaystyle\hphantom{\eqref{228}=}{}-\frac{1}{2}\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)^{2}\Psi(Z)-\theta\sum_{\rho}z_{\rho}\Psi(Z).$

The term indicated by the asterisk is dealt with in complete analogy to the
similarly marked term in (2.16). We thus obtain

$\displaystyle\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}\Psi(Z)+\frac{\hbar^{2}}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\left(-\frac{1}{2}\gamma\left(\frac{[\mathbb{E}_{\rho\sigma},\mathbb{E}_{\sigma\rho}]_{+}}{(z_{\sigma}-z_{\rho})^{2}}\right)+\gamma\big{(}{\rm
e}^{-A^{(1)}-A^{(2)}}\big{)}\left(\frac{\partial_{z_{\rho}}-\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}\right)\right)\Psi\big{(}\widetilde{Z}\big{)}$
$\displaystyle\qquad{}-\frac{1}{2}\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)^{2}\Psi(Z)-\theta\sum_{\rho}z_{\rho}\Psi(Z).$

Hence, putting these all together, we obtain

$\displaystyle\tilde{H}_{\rm
II}\Psi(Q)=\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}\Psi(Z)-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{1}{(z_{\rho}-z_{\sigma})^{2}}\Psi(Z)+\frac{\hbar^{2}}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{\partial_{z_{\rho}}-\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}\Psi(Z)$
$\displaystyle\hphantom{\tilde{H}_{\rm
II}\Psi(Q)=}{}-\frac{1}{2}\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)^{2}\Psi(Z)-\theta\sum_{\rho}z_{\rho}\Psi(Z).$

The equation takes a more convenient form if we apply to the wave function a
gauge transformation of the form

$\Psi(Z)=\exp\left[-\frac{1}{\hbar}\sum_{\alpha}\left(\frac{z_{\alpha}^{3}}{3}+\frac{t}{2}z_{\alpha}\right)\right]\Phi(Z).$

As a result, the Schrödinger equation
$\hbar\partial_{t}\Psi(Z)=\widetilde{H}_{\rm II}\Psi(Z)$ is transformed into
the one with the new Hamiltonian

$\displaystyle\hat{H}_{\rm
II}=\frac{\hbar^{2}}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{\partial_{z_{\sigma}}-\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{1}{\left(z_{\sigma}-z_{\rho}\right)^{2}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}-\hbar\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)\partial_{z_{\rho}}$
$\displaystyle\hphantom{\hat{H}_{\rm II}=}{}+\left(\frac{1}{2}-\theta-\hbar
N\right)\sum_{\rho}z_{\rho}.$ (2.19)

As a result of a similar computation for Calogero–Painlevé I Hamiltonian
operator in (2.3), and each of the Hamiltonian operators in the system (2.7)
(no gauge transformation required), we obtain the following Hamiltonian system
for the quantized Calogero–Painlevé system:

$\displaystyle\tilde{H}_{\rm
I}=\frac{\hbar^{2}}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{\partial_{z_{\sigma}}-\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{1}{\left(z_{\sigma}-z_{\rho}\right)^{2}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}-\sum_{\rho}\left(\frac{z_{\rho}^{3}}{2}+\frac{tz_{\rho}}{4}\right),\\!\\!\\!$
(2.20) $\displaystyle t\tilde{H}_{\rm
III}=\hbar^{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\sigma}^{2}\partial_{z_{\sigma}}-z_{\rho}^{2}\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}+\sum_{\rho}\big{(}\hbar^{2}z_{\rho}^{2}\partial_{z_{\rho}}^{2}-\hbar\big{(}z_{\rho}^{2}-(2\hbar-\theta_{0}+\theta_{1})z_{\rho}-t\big{)}\partial_{z_{\rho}}$
$\displaystyle\hphantom{t\tilde{H}_{\rm III}=}{}-(\hbar
N+\theta_{1})z_{\rho}\big{)}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\rho}^{2}+z_{\sigma}^{2}}{(z_{\sigma}-z_{\rho})^{2}}+\frac{\hbar^{2}N\big{(}1+N^{2}\big{)}}{2},$
(2.21) $\displaystyle\tilde{H}_{\rm
IV}=\hbar^{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\sigma}\partial_{z_{\sigma}}-z_{\rho}\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}+\sum_{\rho}\big{(}\hbar^{2}z_{\rho}\partial_{z_{\rho}}^{2}-\hbar\big{(}z_{\rho}^{2}+tz_{\rho}-\theta_{0}-\hbar\big{)}\partial_{z_{\rho}}\big{)}$
$\displaystyle\hphantom{\tilde{H}_{\rm
IV}}{}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\rho}+z_{\sigma}}{(z_{\sigma}-z_{\rho})^{2}}-(\hbar
N+\theta_{0}+\theta_{1})\sum_{\rho}z_{\rho}-t\hbar N^{2},$ (2.22)
$\displaystyle t\tilde{H}_{\rm
V}=\hbar^{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\sigma}(z_{\sigma}-1)\partial_{z_{\sigma}}-z_{\rho}(z_{\rho}-1)\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}+\hbar^{2}\sum_{\rho}z_{\rho}(z_{\rho}-1)\partial_{z_{\rho}}^{2}$
$\displaystyle\hphantom{t\tilde{H}_{\rm
V}=}{}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\rho}(z_{\rho}-1)+z_{\sigma}(z_{\sigma}-1)}{(z_{\sigma}-z_{\rho})^{2}}$
$\displaystyle\hphantom{t\tilde{H}_{\rm
V}=}{}+\hbar\sum_{\rho}\big{(}tz_{\rho}^{2}+\big{(}2\hbar+\theta_{0}-\theta_{2}-t\big{)}z_{\rho}+\theta_{2}-\hbar\big{)}\partial_{z_{\rho}}+t(\hbar
N+\theta_{0}+\theta_{1})\sum_{\rho}z_{\rho}$
$\displaystyle\hphantom{t\tilde{H}_{\rm
V}=}{}+\left((\theta_{0}-\theta_{2}-t)N^{2}\hbar+\frac{N\hbar^{2}\big{(}1+N^{2}\big{)}}{2}\right),$
(2.23) $\displaystyle t(t-1)\tilde{H}_{\rm
VI}=\hbar^{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\sigma}(z_{\sigma}-1)(z_{\sigma}-t)\partial_{z_{\sigma}}-z_{\rho}(z_{\rho}-1)(z_{\rho}-t)\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}$
$\displaystyle\hphantom{t(t-1)\tilde{H}_{\rm
VI}=}{}+\hbar^{2}\sum_{\rho}z_{\rho}(z_{\rho}-1)(z_{\rho}-t)\partial_{z_{\rho}}^{2}$
$\displaystyle\hphantom{t(t-1)\tilde{H}_{\rm
VI}=}{}+\hbar\sum_{\rho}\big{(}(3\hbar-\theta)z_{\rho}^{2}+\big{(}-\hbar(1+t)+t(\theta_{0}+\theta_{1})+\theta_{0}+\theta_{t}\big{)}z_{\rho}+t(\hbar-\theta_{0})\big{)}\partial_{z_{\rho}}$
$\displaystyle\hphantom{t(t-1)\tilde{H}_{\rm
VI}=}{}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{z_{\rho}(z_{\rho}-1)(z_{\rho}-t)+z_{\sigma}(z_{\sigma}-1)(z_{\sigma}-t)}{(z_{\sigma}-z_{\rho})^{2}}$
$\displaystyle\hphantom{t(t-1)\tilde{H}_{\rm
VI}=}{}+\left(N^{2}\hbar^{2}-\theta
N\hbar-\frac{1}{4}\big{(}k^{2}-\theta^{2}\big{)}+(N-1)\kappa(\kappa+1)\hbar^{2}\right)\sum_{\rho}z_{\rho}$
$\displaystyle\hphantom{t(t-1)\tilde{H}_{\rm
VI}=}{}-\frac{N^{3}\hbar^{2}}{2}+t\hbar N^{2}(\theta_{0}+\theta_{1})+\hbar
N^{2}(\theta_{0}+\theta_{t})-\frac{\hbar^{2}N(N-1)\kappa(\kappa+1)}{2}.$
(2.24)

##  3 Generalization of the quantum Painlevé equations

Nagoya in [22], introduced the Hamiltonian operators corresponding to the
quantum Painlevé equations in the case of a single particle, satisfying the
Schrödinger equation

$\hbar\frac{\partial}{\partial
t}\Phi(z,t)=H_{J}\left(z,\hbar\frac{\partial}{\partial
z},t\right)\Phi(z,t),\qquad J={\rm II},{\rm III},{\rm IV},{\rm V},{\rm VI},$
(3.1)

where Hamiltonian operators $H_{J}$ are obtained from the polynomial
Hamiltonian operators of the Painlevé equations by substituting the operators
$z$, $\hbar\frac{\partial}{\partial z}$ into the canonical coordinates. These
operators are defined as

$\displaystyle H_{\rm
II}=\frac{1}{2}(\hbar\partial_{z})^{2}-\left(z^{2}+\frac{t}{2}\right)\hbar\partial_{z}+az,$
$\displaystyle tH_{\rm
III}=z^{2}(\hbar\partial_{z})^{2}-(z^{2}+bz+t)\hbar\partial_{z}+az,$
$\displaystyle H_{\rm
IV}=z(\hbar\partial_{z})^{2}-\big{(}z^{2}+tz+b\big{)}\hbar\partial_{z}+a(z+t),$
$\displaystyle tH_{\rm
V}=z(z-1)(\hbar\partial_{z})^{2}+\big{(}tz^{2}-(b+c+t)z+b\big{)}\hbar\partial_{z}+a(b+c-a+\hbar+t-tz),$
$\displaystyle t(t-1)H_{\rm
VI}=z(z-1)(z-t)(\hbar\partial_{z})^{2}-\big{(}(a+b)(z-1)(z-t)+cz(z-t)+dz(z-1)\big{)}\hbar\partial_{z}$
$\displaystyle\hphantom{t(t-1)H_{\rm VI}=}{}+(b+c+d+\hbar)a(z-t).$ (3.2)

He showed that the quantum Hamiltonians (3.2) admit special solutions in
integral form when the parameters take certain specific values: in fact, the
wave function $\Phi$ in (3.1) can be taken in the following form

$\Phi_{m}^{J}(z,t)=\int_{\Gamma}\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{i}(z-u_{i})\Theta_{J}(u_{i},t)\mathrm{d}u_{i}.$
(3.3)

Here $\Gamma=\prod_{j}\gamma_{j}$ is a cartesian product of “admissible”
contours $u_{j}\in\gamma_{j}$ that can be chosen on the Riemann surface of the
“master” function $\Theta_{J}$. By this we mean that

  * •

$\Theta_{J}(u_{j})$ is single valued along $\gamma_{j}$;

  * •

$\int_{\gamma_{j}}u_{j}^{k}\Theta_{J}(u_{j})\mathrm{d}u_{j}$ is a convergent
integral and not identically zero (as an expression in $k\in\mathbb{N}$);

  * •

The contours are pairwise non-intersecting if
$\hbar\not\in\frac{1}{2}\mathbb{N}$.

For example for $J={\rm II}$ the contours can be taken as contours starting
from infinity along one of the three directions $\arg(u_{j})=\frac{2\pi}{3}k$,
$k=0,1,2$ and ending at infinity along any of the remaining ones. We could
also take a circle, but then the Cauchy theorem would imply that the integral
$\int u^{\ell}\Theta_{\rm II}(u)\mathrm{d}u$ is zero. The requirement that the
different $\gamma_{j}$’s do not intersect is due to the fact that if
$\hbar\not\in\frac{1}{2}\mathbb{N}$ then the power of the Vandermonde term in
the integrand (3.3) yields a non-single valued function.

The master functions $\Theta_{J}(u_{i},t)$ ($J={\rm II},{\rm III},{\rm
IV},{\rm V},{\rm VI}$) are the weight functions defined below:

$\displaystyle\Theta_{\rm
II}=\exp\left(-\left(u_{i}t+\frac{2}{3}u_{i}^{3}\right)\right),$
$\displaystyle\Theta_{\rm
III}=u_{i}^{-b-1}\exp\left(\frac{t}{u_{i}}-u_{i}\right),$
$\displaystyle\Theta_{\rm
IV}=u_{i}^{-b-1}\exp\left(-\left(u_{i}t+\frac{u_{i}^{2}}{2}\right)\right),$
$\displaystyle\Theta_{\rm V}=u_{i}^{-b-1}(1-u_{i})^{-c-1}\exp(u_{i}t),$
$\displaystyle\Theta_{\rm VI}=u_{i}^{-a-b-1}(1-u_{i})^{-c-1}(t-u_{i})^{-d}.$
(3.4)

With the positions (3.4) and formula (3.3) the functions $\Phi_{m}^{J}$
satisfy the Schrödinger equations (3.1) provided that the parameters $a$, $b$,
$c$, $d$ satisfy the following condition

$\begin{cases}a=m\hbar\quad\text{and}\quad b+c+d=(m-1)\hbar,&J={\rm VI},\\\
a=m\hbar,&J={\rm II},{\rm III},{\rm IV},{\rm V}.\end{cases}$

We want to generalize this result of Nagoya’s to our multi-particle quantum
Hamiltonians (2.19), (2.21)–(2.24) by providing a multi-particle extension of
the integral formulæ (3.3).

###  3.1 Integral representations for the quantum Painlevé–Calogero
Schrödinger
wave functions

We now explain the general approach behind the extension of Nagoya’s formulæ.
We start from an Ansatz of the form

$\Phi(\vec{z};t)=\int_{\Gamma}\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{\rho=1}^{N}\prod_{i=1}^{m}(z_{\rho}-u_{i})\Theta_{J}(u_{i})\mathrm{d}u_{i}.$
(3.5)

Observe that, similarly to Nagoya’s result, these are polynomials in the
$z_{\rho}$’s of total degree $Nm$ and of degree $m$ in each of the variables
$z_{\rho}$. With the Ansatz (3.5) in place, we verify by a direct calculation
on a case-by-case basis, that they satisfy a multi-variate generalization of
(3.1) and identify the corresponding Hamiltonian operator. The result of these
computations, whose details are reported in the appendix section, are
summarized in the following theorem (for readability, the range of greek
indices is assumed to be $1,\dots,N$ without explicit mention).

######  Theorem 3.1.

The equation (3.5), with $\Theta_{J}$ given by (3.4), is a solution to the
multi-variable version of Schrödinger equation (3.1), where the Hamiltonians
$H_{J}$, $J={\rm II},{\rm III},{\rm IV},{\rm V},{\rm VI}$ are given by the
following operators:

$\displaystyle H_{\rm
II}=\frac{\hbar}{2}\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{\partial_{z_{\rho}}-\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}-\hbar\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)\partial_{z_{\rho}}+m\hbar\sum_{\rho}z_{\rho},$
(3.6) $\displaystyle tH_{\rm
III}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{z_{\rho}^{2}\partial_{z_{\rho}}-z_{\sigma}^{2}\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}+\sum_{\rho}\big{(}\hbar^{2}z_{\rho}^{2}\partial_{z_{\rho}}^{2}-\hbar\big{(}z_{\rho}^{2}+(b+N-1)z_{\rho}+t\big{)}\partial_{z_{\rho}}+m\hbar
z_{\rho}\big{)},$ (3.7) $\displaystyle H_{\rm
IV}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{z_{\rho}\partial_{z_{\rho}}-z_{\sigma}\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}+\sum_{\rho}\big{(}\hbar^{2}z_{\rho}\partial_{z_{\rho}}^{2}-\hbar\big{(}z_{\rho}^{2}+tz_{\rho}+b\big{)}\partial_{z_{\rho}}+m\hbar
z_{\rho}\big{)}+\hbar Nmt,$ (3.8) $\displaystyle tH_{\rm
V}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{z_{\rho}(z_{\rho}-1)\partial_{z_{\rho}}-z_{\sigma}(z_{\sigma}-1)\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}+\hbar
Nm\big{(}b+c+t-\hbar(m-1)-N+1\big{)}$ $\displaystyle\hphantom{tH_{\rm
V}=}{}+\sum_{\rho}\big{(}\hbar^{2}z_{\rho}(z_{\rho}-1)\partial_{z_{\rho}}^{2}+\hbar\big{(}tz_{\rho}^{2}-(b+c+t)z_{\rho}+b\big{)}\partial_{z_{\rho}}-m\hbar
tz_{\rho}\big{)},$ (3.9) $\displaystyle t(t-1)H_{\rm
VI}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{z_{\rho}(z_{\rho}-1)(z_{\rho}-t)\partial_{z_{\rho}}-z_{\sigma}(z_{\sigma}-1)(z_{\sigma}-t)\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}$
$\displaystyle\hphantom{t(t-1)H_{\rm
VI}=}{}+\sum_{\rho}\hbar^{2}z_{\rho}(z_{\rho}-1)(z_{\rho}-t)\partial_{z_{\rho}}^{2}$
$\displaystyle\hphantom{t(t-1)H_{\rm
VI}=}{}-\hbar\sum_{\rho}\left((a+b)(z_{\rho}-1)(z_{\rho}-t)+cz_{\rho}(z_{\rho}-t)+(d+N-1)z_{\rho}(z_{\rho}-1)\right)\partial_{z_{\rho}}$
$\displaystyle\hphantom{t(t-1)H_{\rm VI}=}{}-\hbar m(N-1-\hbar
m)\sum_{\rho}z_{\rho}-\hbar mN(\hbar m+1-N)t.$ (3.10)

Quantum Calogero Hamiltonians and integral solutions. We now examine the
relationship between the Hamiltonians (differential operators) (3.6)–(3.10),
whose integral solutions are given (3.5) and the quantum Calogero Hamiltonians
(2.19), (2.21)–(2.24) constructed by canonical quantization of the non-
commutative Hamiltonians of the classical isomonodromic noncommutative
equations of [2]. By way of this identification, we identify the parameter
$\kappa$ of the quantum radial reduction up to an appropriate gauge
transformation of the Hamiltonians. At the end of this computation, we can see
that the Hamiltonians (2.19), (2.21)–(2.24) reduce to (3.6)–(3.10) for
particular choice of the parameters. The results are summarized in Table 1.
Consider for example (2.19) and (3.6):

$\displaystyle\tilde{H}_{\rm
II}=\frac{\hbar^{2}}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{\partial_{z_{\sigma}}-\partial_{z_{\rho}}}{z_{\sigma}-z_{\rho}}-\frac{\hbar^{2}\kappa(\kappa+1)}{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho\neq\sigma\end{subarray}}\frac{1}{\left(z_{\sigma}-z_{\rho}\right)^{2}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}-\hbar\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)\partial_{z_{\rho}}$
$\displaystyle\hphantom{\tilde{H}_{\rm II}=}{}+\left(\frac{1}{2}-\theta-\hbar
N\right)\sum_{\rho}z_{\rho},$ $\displaystyle H_{\rm
II}=\frac{\hbar}{2}\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{\partial_{z_{\rho}}-\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{z_{\rho}}^{2}-\hbar\sum_{\rho}\left(z_{\rho}^{2}+\frac{t}{2}\right)\partial_{z_{\rho}}+m\hbar\sum_{\rho}z_{\rho}.$

The first observation is that in the two Hamiltonians the second and first-
order differential parts have different powers of $\hbar$; they coincide only
for $\hbar=1$ (we exclude the trivial case $\hbar=0$). We will comment on the
$\hbar=1$ case later on.

The second observation is about the Calogero-like potential term in (2.19)
which is absent in (3.6); the term disappears for $\kappa=0,-1$. These values
mean that the ${\rm GL}_{n}$ representation in $\mathbb{V}$ is the trivial one
in the radial quantization scheme.

Then the other parameters in the equations can be easily matched and then we
need $\theta=\frac{1}{2}-N-m$.

A more general family of matching between the two sets of operators is
obtained by gauging the Hamiltonians (3.6)–(3.10) by means of an appropriate
power of the Weyl denominator, namely, the Vandermonde determinant. We
indicate this in the following lemma:

######  Lemma 3.2.

Let $\Delta(\vec{z})=\prod_{1\leq\alpha<\beta\leq N}(z_{\alpha}-z_{\beta})$ be
the Vandermonde polynomial in $\vec{z}$. Then the action of the quantum
Hamiltonian operator $H_{\rm II}$ on the generalized wave functions, is
equivalent to the action of $\Delta^{-R}\widetilde{H}_{\rm II}\Delta^{R}$:

$\displaystyle\Delta^{-R}\widetilde{H}_{\rm II}\Delta^{R}\Phi(Z)=H_{\rm
II}\Phi(Z)$ (3.11)

for $R$ and the scalar $\kappa$ determined as one of the two choices below:

$\displaystyle\left(R=\frac{1}{\hbar}-1,\,\kappa=\frac{1}{\hbar}-1\right),\qquad\left(R=\frac{1}{\hbar}-1,\,\kappa=-\frac{1}{\hbar}\right),\qquad\hbar\neq
0.$

Under this circumstances, the parameter $\theta$ is determined as $($in either
case$)$:

$\displaystyle\theta=\hbar(1-m)+N(1-2\hbar)-\frac{1}{2}.$

###### Proof.

The proof is obtained by a direct computation of the equations in (3.11). ∎

Lemma 3.2 is actually the manifestation of a more general result which is
contained in the following theorem:

######  Theorem 3.3.

Define the two sequences of differential operators

$\displaystyle
H_{a}:=\hbar^{2}\sum_{\rho}z_{\rho}^{a}\partial_{z_{\rho}}^{2}+2\hbar\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho<\sigma\end{subarray}}\frac{z_{\rho}^{a}\partial_{z_{\rho}}-z_{\sigma}^{a}\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}},$
$\displaystyle\widetilde{H}_{a}:=\hbar^{2}\sum_{\rho}z_{\rho}^{a}\partial_{z_{\rho}}^{2}+2\hbar^{2}\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho<\sigma\end{subarray}}\frac{z_{\rho}^{a}\partial_{z_{\rho}}-z_{\sigma}^{a}\partial_{z_{\sigma}}}{z_{\rho}-z_{\sigma}}-\hbar^{2}\kappa(\kappa+1)\sum_{\begin{subarray}{c}\rho,\sigma\\\
\rho<\sigma\end{subarray}}\frac{z_{\rho}^{a}+z_{\sigma}^{a}}{(z_{\rho}-z_{\sigma})^{2}}$
$\displaystyle\hphantom{\widetilde{H}_{a}:=}{}+\begin{cases}0,&a=0,1,\\\
\dfrac{N(N-1)(N-2)}{3},&a=2,\\\
\displaystyle(N-1)(N-2)\sum_{\rho}z_{\rho},&a=3.\end{cases}$

Then we have the following identity

$H_{a}=\Delta^{-R}\circ\widetilde{H}_{a}\circ\Delta^{R}$

provided that $R=\frac{1}{\hbar}-1$ and $\kappa=\frac{1}{\hbar}-1$ or
$\kappa=-\frac{1}{\hbar}$.

We do not report here the straightforward proof. Using Theorem 3.3 in the
various cases of the Hamiltonian Calogero–Painlevé operators and direct
computations, allows to express the Hamiltonians (2.19), (2.21)–(2.24) as
special cases of (3.6)–(3.10) and allows us to identify the parameters
$\theta_{0}$, $\theta_{1}$, $\theta_{2}$, $\theta_{t}$, $\theta$, $k$, $a$,
$b$, $c$ and $d$. The result of the computations is summarized in Table 1.

Table 1: The correspondence of parameters between the differential operators (3.6)–(3.10) and (2.19), (2.21)–(2.24) with the relationship for the parameters. | $H\Psi(Z)=\widetilde{H}\Psi(Z)$ | $\Delta^{-R}\widetilde{H}\Delta^{R}\Psi(Z)=H\Psi(Z)$
---|---|---
Calogero–Painlevé II | $\begin{matrix}\kappa=0,\quad\hbar=1,\\\ \theta=\frac{1}{2}-N-m\end{matrix}$ | $\begin{matrix}\kappa=\frac{1}{\hbar}-1\quad\hbox{or}\quad\kappa=-\frac{1}{\hbar},\quad R=\frac{1}{\hbar}-1,\\\ \theta=\hbar(1-m)+N(1-2\hbar)-\frac{1}{2}\end{matrix}$
Calogero–Painlevé III | $\begin{matrix}\kappa=0,\quad\hbar=1,\\\ \theta_{0}=b-m+1,\\\ \theta_{1}=-N-m\end{matrix}$ | $\begin{matrix}\kappa=\frac{1}{\hbar}-1\quad\hbox{or}\quad\kappa=-\frac{1}{\hbar},\quad R=\frac{1}{\hbar}-1,\\\ \theta_{0}=b+\hbar(1-m),\\\ \theta_{1}=-\hbar(m+1)-N+1\end{matrix}$
Calogero–Painlevé IV | $\begin{matrix}\kappa=0,\quad\hbar=1,\\\ \theta_{0}=-b-1,\\\ \theta_{1}=b+1-m-N\end{matrix}$ | $\begin{matrix}\kappa=\frac{1}{\hbar}-1\quad\hbox{or}\quad\kappa=-\frac{1}{\hbar},\quad R=\frac{1}{\hbar}-1,\\\ \theta_{0}=-b-\hbar,\\\ \theta_{1}=b+1-N-m\hbar\end{matrix}$
Calogero–Painlevé V | $\begin{matrix}\kappa=0,\quad\hbar=1,\\\ \theta_{0}=-c-1,\\\ \theta_{1}=-N-m+c+1,\\\ \theta_{2}=b+1\end{matrix}$ | $\begin{matrix}\kappa=\frac{1}{\hbar}-1\quad\hbox{or}\quad\kappa=-\frac{1}{\hbar},\quad R=\frac{1}{\hbar}-1,\\\ \theta_{0}=-c-\hbar,\\\ \theta_{1}=c+1-N-m\hbar,\\\ \theta_{2}=b+\hbar\end{matrix}$
Calogero–Painlevé VI | $\begin{matrix}\kappa=0,\quad\hbar=1,\\\ \theta_{0}=a+b+1,\\\ \theta_{1}=c+1,\\\ \theta_{t}=d+N,\\\ k=\pm\sqrt{A}\\\ A=(\theta-2N)^{2}+4m(N-1-m)\end{matrix}$ | $\begin{matrix}\kappa=\frac{1}{\hbar}-1\quad\hbox{or}\quad\kappa=-\frac{1}{\hbar},\quad R=\frac{1}{\hbar}-1,\\\ \theta_{0}=a+b+\hbar,\\\ \theta_{1}=c+\hbar,\\\ \theta_{t}=d+N+\hbar-1,\\\ k=\pm\sqrt{B}\\\ B=(\theta-2N\hbar)^{2}+4\hbar m(N-1-\hbar m)\\\ {}+4(N-1)(1-\hbar)+N(N-1)(1-\hbar)(3\hbar-\theta)\end{matrix}$

With these values of the parameters then quantum Calogero–Painlevé and
generalized quantum Painlevé equations II–VI yield the same Hamiltonians, and
the integral representation (3.5) provides solutions of the corresponding
Schrödinger equations (3.1).

We now briefly comment on the value of the Heisenberg constant $\hbar=1$:
observing the original integral representation of Nagoya (3.3) and the
generalized one (3.5), we see that for $\hbar=1$ the integrand contains the
square of the Vandermonde determinant of the variables $u_{j}$. This type of
expression is very familiar in the context of random matrices [21]: it is the
Jacobian of the change of variables from the Lebesgue measure on Hermitian
matrices (or normal matrices) to the unitary-radial coordinates. Specifically,
if $M$ is a Hermitean matrix of size $m\times m$, we write it as $M=VDV$ with
$D=\operatorname{diag}(u_{1},\dots,u_{m})$ and $V\in U(m,\mathbb{C})$, then
the Lebesque measure (up to inessential multiplicative constant) is

$\mathrm{d}M=\Delta(u)^{2}\mathrm{d}V\prod_{j}\mathrm{d}u_{j},$

where $\Delta(u)=\prod_{i<j}(u_{i}-u_{j})$. This expression is also valid if
the $u_{j}$’s are allowed to take complex values along specified curves (but
now $\mathrm{d}M$ is the measure on $m\times m$ normal matrices). This allows
us to rewrite the integral formulæ as matrix integrals; for example for
Painlevé II we have

$\displaystyle\Phi_{m}^{\rm
II}(\vec{z},t)=\int\prod_{\rho}\det(z_{\rho}-M){\rm
e}^{-\operatorname{Tr}\left(\frac{2}{3}M^{3}+tM\right)}\mathrm{d}M,$ (3.12)

which expresses the wave function as the expectation value of the product of
characteristic polynomials. Similar expressions hold for the other cases.
Therefore it appears that the class of generalized Nagoya solutions (3.5) and
the wave-functions of the quantum Calogero–Painlevé Hamiltonians intersect on
the class of solutions that are related to matrix integrals of the form (3.12)
only (and their similar expressions for the other master functions (3.4)).

We conclude by observing that we can write (3.12) in terms of the matrix
variable $Q$ (with eigenvalues $z_{1},\dots,z_{N}$) as follows

$\Phi_{m}^{\rm
II}(Q,t)=\int\prod_{\rho}\det(Q\otimes\mathbf{1}_{m}-\mathbf{1}_{N}\otimes
M){\rm e}^{-\operatorname{Tr}\left(\frac{2}{3}M^{3}+tM\right)}\mathrm{d}M.$

For different values of $\hbar$ we need to use the substitution indicated in
Theorem 3.3; the resulting integral representation is what is known as a
“$\beta$-integral”. Thus, we obtain the following

######  Theorem 3.4.

Let $\Phi_{J}(\vec{z};t)$ be the wave function (3.5) solving the Schrödinger
equations with Hamiltonians (3.6)–(3.10). Then
$\Psi_{J}(\vec{z},t)=\Delta^{\frac{1}{\hbar}-1}\Phi_{J}(\vec{z};t)$ is a
solution of the corresponding Painlevé $J$ Hamiltonians $(J={\rm
II},\dots,{\rm VI})$ (2.19), (2.21)–(2.24), i.e.,

$\Psi_{J}(\vec{z};t)=\Delta(\vec{z})^{\frac{1}{\hbar}-1}\int\Delta(\vec{u})^{2\hbar}\prod_{\rho,j}(z_{\rho}-u_{j})\Theta_{J}(u_{j},t)\mathrm{d}u_{i}$

give solutions to the multi-particle quantized Calogero–Painlevé equations
with Hamiltonians (2.19), (2.21)–(2.24) for
$\kappa=-\frac{1}{\hbar},\frac{1}{\hbar}-1$ $($respectively$)$ and the values
of parameters indicated in Table 1.

##  4 Conclusions

In recent years, there have been different approaches to “quantum Painlevé
equations” using as starting point the linear differential equation of rank
$2$ classically associated to the Painlevé equations, or the theory of
topological recursion associated to semiclassical spectral curves.

In [27], the authors start from the system of linear equations (Lax system)
associated to the Painlevé equations I–VI written in the form

$\begin{cases}\partial_{z}\Psi=U(z,t)\Psi,\\\
\partial_{t}\Psi=V(z,t)\Psi.\end{cases}$ (4.1)

After applying a suitable change of variables and gauge transformation, they
obtain a pair of compatible PDEs for a scalar wave function $\psi$ (obtained
from the $(1,1)$ entry of the matrix $\Psi$)

$\begin{cases}\big{(}\frac{1}{2}\partial_{z}^{2}-\frac{1}{2}(\partial_{z}\log
b)\partial_{z}+W(z,t)\big{)}\psi=0,\\\
\partial_{t}\psi=\big{(}\frac{1}{2}\partial_{z}^{2}+\mathcal{U}(z,t)\big{)}\psi,\end{cases}$
(4.2)

where $W$ and $\mathcal{U}$ are described explicitly in terms of the entries
of the matrices $U$, $V$. The first equation in (4.2) has apparent
singularities but otherwise exhibits the same (generalized) monodromy
associated to (4.1) (in $\mathbb{P}{\rm SL}_{2}$), while the second equation
describes the isomonodromic deformation of the former and is presented in the
form of a non-stationary Schrödinger equation with imaginary time.

The Hamiltonian operators corresponding to each Painlevé equation I–VI from
the second equation of the system (4.2) are, in particular, a natural
quantization of those corresponding to the Calogero-like Painlevé equations
(those obtained from the first equation of the system (4.2)). These operators
that are called the quantum Calogero–Painlevé Hamiltonian system, are obtained
in single variable representation. The question then arises as to whether a
similar description is possible in the multi-particle case; the naïve approach
of considering the matrices as $2\times 2$ blocks does not lead to equations
of the same type as (4.2). The quantization of the Hamiltonians from [2] does
not seem to be the direct analog of (4.2) because it is an equation where the
wave equation plays the role rather of the (scalar) “quantum tau function”.

Also, in [9], the authors use the topological recursion on spectral curves of
different genuses which results in wave functions that satisfy a family of
partial differential equations. In fact, these PDEs are the quantization of
the original spectral curves. As an application of their theorem, they
introduce a system of PDEs corresponding to Painlevé transcendents whose
assigned Hamiltonian system has significant similarities to the system of
Hamiltonian operators that we introduced in this paper for the quantum
Calogero–Painlevé system.

Finally, we comment on the possible relationship with equations of the
Knizhnik–Zamolodchikov (KZ) type. For the single-particle case, in [22], H.
Nagoya provides a representation-theoretic correspondence between the
Schrödinger equation for quantum Painlevé VI (single-particle) and the
Knizhnik–Zamolodchikov (KZ) equation, and between the Schrödinger equation for
quantum Painlevé II–V (single-particle) and the confluent KZ equations that
are defined in [17]. These correspondences are proved directly by showing
relations between the integral representations for the solutions to the
quantum Painlevé equations and solutions to the (confluent) KZ equations. In
this case, the present paper should play a similar role for
$\mathfrak{sl}_{n}$ special solutions of the KZ equations. We plan to address
this possible relationship in a subsequent publication.

##  Appendix A Proofs

The starting point of all the following computations is to take the integral
representation (3.5)

$\Phi(\vec{z};t)=\int_{\Gamma}\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}(z_{\rho}-u_{i})\Theta_{J}(u_{i})\mathrm{d}u_{i},$

and apply to it the direct sum of the second-order parts in the quantum
Hamiltonians (3.2).

For the sake of simplicity of the notation in the following computations, We
denote by $\langle F(u_{1},\dots,u_{m})\rangle$ the un-normalized expectation
value as follows

$\langle F(u_{1},\dots,u_{m})\rangle:=\int_{\Gamma}\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{i}\Theta_{J}(u_{i})F(u_{1},\dots,u_{m})\mathrm{d}u_{i}.$

###  A.1 Quantum Painlevé II

Define $P(\vec{z})=\prod_{\rho,i}({z}_{\rho}-u_{i})$ and $\Delta=\prod_{1\leq
i<j\leq m}(u_{i}-u_{j})$, then

$\displaystyle\sum_{\rho}\partial_{{z}_{\rho}}^{2}P=\sum_{\rho}P\sum_{i\neq
j}\frac{1}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\hphantom{\sum_{\rho}\partial_{{z}_{\rho}}^{2}P}{}=\sum_{\rho}P\sum_{i\neq
j}\left(\frac{1}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{1}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right).$

This yields

$\displaystyle\hbar^{2}\sum_{\rho}\partial_{{z}_{\rho}}^{2}\Phi=\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\left(\frac{1}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{1}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right)\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}\partial_{{z}_{\rho}}^{2}\Phi}{}=2\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\frac{1}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}\partial_{{z}_{\rho}}^{2}\Phi}{}=\hbar\int\sum_{i}\partial_{u_{i}}\big{(}\Delta^{2\hbar}\big{)}\sum_{\rho}\frac{1}{{z}_{\rho}-u_{i}}P\prod_{k}\Theta(u_{k})\mathrm{d}u_{i}.$

We now use integration by parts in the integrand and obtain

$\displaystyle=-\hbar\int\Delta^{2\hbar}\sum_{\rho}\sum_{i}\partial_{u_{i}}\bigg{(}\frac{1}{{z}_{\rho}-u_{i}}P\prod_{k}\Theta(u_{k})\bigg{)}\mathrm{d}u_{i}$
$\displaystyle=-\hbar\int\Delta^{2\hbar}\sum_{\rho}\sum_{i}\left(\frac{P}{({z}_{\rho}-u_{i})^{2}}+\frac{P_{u_{i}}}{{z}_{\rho}-u_{i}}-\frac{2u_{i}^{2}+t}{{z}_{\rho}-u_{i}}P\right)\prod_{k}\Theta(u_{k})\mathrm{d}u_{i}.$

After simplifying we obtain

$\displaystyle\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{{z}_{\rho}}^{2}\Phi=-\frac{\hbar}{2}\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{\partial_{{z}_{\rho}}-\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi+\hbar\sum_{\rho}\left({z}_{\rho}^{2}+\frac{t}{2}\right)\partial_{{z}_{\rho}}\Phi-\hbar
m\sum_{\rho}{z}_{\rho}\Phi+\hbar N\partial_{t}\Phi.$

Rearranging the terms appropriately, we obtain the Schrödinger equation with
the Hamiltonian (3.6):

$\displaystyle H_{\rm
II}=\frac{\hbar}{2}\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{\partial_{{z}_{\rho}}-\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}+\frac{\hbar^{2}}{2}\sum_{\rho}\partial_{{z}_{\rho}}^{2}-\hbar\sum_{\rho}\left({z}_{\rho}^{2}+\frac{t}{2}\right)\partial_{{z}_{\rho}}+m\hbar\sum_{\rho}{z}_{\rho}.$

###  A.2 Quantum Painlevé III

The initial setup matches the previous case, except that we need to consider
the direct sum of the operators ${z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}$
acting on the wave function. To this end, we observe that

$\displaystyle\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}P({\vec{z}})=\sum_{\rho}P\sum_{i\neq
j}\frac{{z}_{\rho}^{2}}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\hphantom{\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}P({\vec{z}})}{}=\sum_{\rho}P\sum_{i\neq
j}\left(1+\frac{u_{i}^{2}}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}^{2}}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right).$

This yields

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\left(1+\frac{u_{i}^{2}}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}^{2}}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right)\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi}{}=\hbar^{2}Nm(m-1)\Phi+\hbar
Nm\Phi+\hbar N^{2}m\Phi-\hbar
N\sum_{\sigma}{z}_{\sigma}\partial_{{z}_{\sigma}}\Phi-\hbar\sum_{\rho}{z}_{\rho}\sum_{\sigma}\partial_{{z}_{\sigma}}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=}{}+\hbar\sum_{\rho}({z}_{\rho}^{2}+b{z}_{\rho}+t)\partial_{{z}_{\rho}}\Phi-\hbar
bNm\Phi-\hbar Nm\Phi+\hbar\sum_{\rho}\partial_{{z}_{\rho}}\Phi-\hbar
m\sum_{\rho}{z}_{\rho}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=}{}-\hbar
N\bigg{\langle}\\!\sum_{i}u_{i}P\\!\bigg{\rangle}-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}^{2}\partial_{{z}_{\rho}}-{z}_{\sigma}^{2}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi+\hbar\sum_{\rho}{z}_{\rho}\sum_{\sigma}\partial_{{z}_{\sigma}}\Phi-\hbar\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=}{}+\hbar(N-1)\sum_{\sigma}{z}_{\sigma}\partial_{{z}_{\sigma}}\Phi.$

Rearranging the terms we obtain the following expression

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}^{2}\partial_{{z}_{\rho}}-{z}_{\sigma}^{2}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi+\hbar\sum_{\rho}\big{(}{z}_{\rho}^{2}+b{z}_{\rho}+t\big{)}\partial_{{z}_{\rho}}\Phi-m\hbar\sum_{\rho}{z}_{\rho}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=}{}+t\hbar
N\partial_{t}\Phi+\hbar^{2}Nm(m-1)\Phi+\hbar N^{2}m\Phi-
bNm\hbar\Phi-\hbar\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}\Phi=}{}-\hbar
N\bigg{\langle}\sum_{i}\left(\frac{t}{u_{i}}+u_{i}\right)P\bigg{\rangle}.$
(A.1)

In order to handle the remaining expectation value we need to derive some
further identities: consider the Euler differential operator

$\mathbb{E}:=\sum_{i}u_{i}\partial_{u_{i}}.$

Applying this operator $\mathbb{E}$ to the integrand of $\Phi_{m}^{\rm III}$,

$\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-b-1}{\rm
e}^{(\frac{t}{u_{i}}-u_{i})}$

gives the following expression

$\displaystyle\sum_{i}u_{i}\partial_{u_{i}}\bigg{(}\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-b-1}{\rm
e}^{(\frac{t}{u_{i}}-u_{i})}\bigg{)}$
$\displaystyle=\sum_{i}u_{i}\bigg{[}\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\bigg{(}(2\hbar)\sum_{i\neq
j}\frac{1}{u_{i}-u_{j}}\bigg{)}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-b-1}{\rm
e}^{(\frac{t}{u_{i}}-u_{i})}$
$\displaystyle\quad{}+\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})\bigg{(}\sum_{\rho}\frac{-1}{{z}_{\rho}-u_{i}}\bigg{)}u_{i}^{-b-1}{\rm
e}^{(\frac{t}{u_{i}}-u_{i})}$
$\displaystyle\quad{}+\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-b-1}{\rm
e}^{(\frac{t}{u_{i}}-u_{i})}\left(\frac{-b-1}{u_{i}}-\frac{t}{u_{i}^{2}}-1\right)\bigg{]}$
$\displaystyle=\sum_{i}u_{i}\bigg{[}2\hbar\sum_{i\neq
j}\frac{1}{u_{i}-u_{j}}-\sum_{\rho}\frac{1}{{z}_{\rho}-u_{i}}-\frac{b+1}{u_{i}}-\frac{t}{u_{i}^{2}}-1\bigg{]}\Delta^{2\hbar}P\prod_{i}\Theta(u_{i})$
$\displaystyle=\Delta^{2\hbar}\bigg{[}\hbar
m(m-1)P+NmP-\sum_{\rho}{z}_{\rho}\partial_{z_{\rho}}P-(b+1)mP-\sum_{i}\left(\frac{t}{u_{i}}+u_{i}\right)P\bigg{]}\prod_{i}\Theta(u_{i}).$

Then we observe that

$\displaystyle\int\sum_{j}u_{j}\partial_{u_{j}}\bigg{(}\Delta^{2\hbar}P\prod_{i}\Theta(u_{i})\mathrm{d}u_{i}\bigg{)}=\int\sum_{j}\partial_{u_{j}}\bigg{(}u_{j}\Delta^{2\hbar}P\prod_{i}\Theta(u_{i})\mathrm{d}u_{i}\bigg{)}-m\Phi(z),$

and the first integral is zero because the integrand is a divergence of a
vector. Therefore,

$\displaystyle\underbrace{\int\mathbb{E}\bigg{(}\Delta^{2\hbar}P\prod_{i}\Theta(u_{i})\mathrm{d}u_{i}\bigg{)}}_{=-m\Phi}$
$\displaystyle\qquad{}=\hbar
m(m-1)\Phi+Nm\Phi-\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}\Phi-(b+1)m\Phi-\bigg{\langle}\sum_{i}\left(\frac{t}{u_{i}}+u_{i}\right)P\bigg{\rangle}$
$\displaystyle\Longrightarrow\quad\bigg{\langle}\sum_{i}(\frac{t}{u_{i}}+u_{i})P\bigg{\rangle}=\hbar
m(m-1)\Phi+Nm\Phi-\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}\Phi-b\Phi.$

Substituting the left side into the equation (A.1) results in the following
conclusion

$\displaystyle t\hbar
N\partial_{t}\Phi=\sum_{\rho}\big{(}\hbar^{2}{z}_{\rho}^{2}\partial_{z_{\rho}}^{2}-\hbar\big{(}{z}_{\rho}^{2}+(b+N-1){z}_{\rho}+t\big{)}\partial_{{z}_{\rho}}+m\hbar{z}_{\rho}\big{)}\Phi+\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}^{2}\partial_{{z}_{\rho}}-{z}_{\sigma}^{2}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi.$

Hence, the general Hamiltonian operator for quantum Painlevé III equation is
given by

$\displaystyle tH_{\rm
III}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}^{2}\partial_{{z}_{\rho}}-{z}_{\sigma}^{2}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}+\sum_{\rho}\big{(}\hbar^{2}{z}_{\rho}^{2}\partial_{{z}_{\rho}}^{2}-\hbar\big{(}{z}_{\rho}^{2}+(b+N-1){z}_{\rho}+t\big{)}\partial_{{z}_{\rho}}+m\hbar{z}_{\rho}\big{)}.$

###  A.3 Quantum Painlevé IV

We follow the same general scheme and consider the sum of the
$z_{\rho}\partial_{z\rho}^{2}$ term applied to $P$:

$\displaystyle\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}P=\sum_{\rho}{z}_{\rho}P\sum_{i\neq
j}\frac{1}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\hphantom{\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}P}{}=\sum_{\rho}P\sum_{i\neq
j}\left(\frac{u_{i}}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right).$

This yields

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi=\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\left(\frac{u_{i}}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right)\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi}{}=2\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\frac{u_{i}}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi}{}=\hbar\int\sum_{i}\partial_{u_{i}}\big{(}\Delta^{2\hbar}\big{)}\sum_{\rho}\frac{u_{i}}{{z}_{\rho}-u_{i}}P\prod_{k}\Theta(u_{k})\mathrm{d}u_{i}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi}{}=-\hbar\int\Delta^{2\hbar}\sum_{\rho}\sum_{i}\partial_{u_{i}}\bigg{(}\left(-1+\frac{{z}_{\rho}}{{z}_{\rho}-u_{i}}\right)P\prod_{k}\Theta(u_{k})\bigg{)}\mathrm{d}u_{i},$

which gives

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi=-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}\partial_{{z}_{\rho}}-{z}_{\sigma}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi+\hbar\sum_{\rho}\big{(}{z}_{\rho}^{2}+t{z}_{\rho}+b\big{)}\partial_{{z}_{\rho}}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}^{2}\Phi=}{}-\hbar
Nmt\Phi+\hbar N\partial_{t}\Phi-\hbar m\sum_{\rho}{z}_{\rho}\Phi.$

Rearranging the terms in the above expression, the general Hamiltonian
operator for quantum Painlevé IV equation is given by

$\displaystyle H_{\rm
IV}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}\partial_{{z}_{\rho}}-{z}_{\sigma}\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}+\sum_{\rho}\big{(}\hbar^{2}{z}_{\rho}\partial_{{z}_{\rho}}^{2}-\hbar\big{(}{z}_{\rho}^{2}+t{z}_{\rho}+b\big{)}\partial_{{z}_{\rho}}+m\hbar{z}_{\rho}\big{)}+\hbar
Nmt.$

###  A.4 Quantum Painlevé V

We start from the same set up as previous operators

$\displaystyle\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}P=\sum_{\rho}P\sum_{i\neq
j}\frac{{z}_{\rho}({z}_{\rho}-1)}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\hphantom{\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}P}{}=\sum_{\rho}P\sum_{i\neq
j}\left(1+\frac{u_{i}(u_{i}-1)}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}(u_{j}-1)}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right).$

This yields

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi=\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\left(1+\frac{u_{i}(u_{i}-1)}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}-\frac{u_{j}(u_{j}-1)}{({z}_{\rho}-u_{j})(u_{i}-u_{j})}\right)\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi}{}=\hbar^{2}Nm(m-1)\Phi+2\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\frac{u_{i}(u_{i}-1)}{({z}_{\rho}-u_{i})(u_{i}-u_{j})}\bigg{\rangle}$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi}{}=\hbar^{2}Nm(m-1)\Phi+\hbar\int\sum_{i}\partial_{u_{i}}\big{(}\Delta^{2\hbar}\big{)}\sum_{\rho}\frac{u_{i}(u_{i}-1)}{{z}_{\rho}-u_{i}}P\prod_{k}\Theta(u_{k})\mathrm{d}u_{i},$
$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi=-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}-{z}_{\sigma}({z}_{\sigma}-1)\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi=}{}-\hbar\sum_{\rho}\big{(}t{z}_{\rho}^{2}-(b+c+t){z}_{\rho}+b\big{)}\partial_{{z}_{\rho}}\Phi-\hbar
Nm(b+c+t)\Phi$
$\displaystyle\hphantom{\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}\Phi=}{}+m\hbar
t\sum_{\rho}{z}_{\rho}\Phi+\hbar Nt\partial_{t}\Phi+\hbar^{2}Nm(m-1)\Phi+\hbar
Nm(N-1)\Phi.$

Therefore, the general Hamiltonian operator for quantum Painlevé V equation is
given by

$\displaystyle tH_{\rm
V}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}-{z}_{\sigma}({z}_{\sigma}-1)\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}$
$\displaystyle\hphantom{tH_{\rm
V}=}{}+\sum_{\rho}\big{(}\hbar^{2}{z}_{\rho}({z}_{\rho}-1)\partial_{{z}_{\rho}}^{2}+\hbar\big{(}t{z}_{\rho}^{2}-(b+c+t){z}_{\rho}+b\big{)}\partial_{{z}_{\rho}}-m\hbar
t{z}_{\rho}\big{)}$ $\displaystyle\hphantom{tH_{\rm V}=}{}+\hbar
Nm(b+c+t)-\hbar^{2}Nm(m-1)-\hbar Nm(N-1).$

###  A.5 Quantum Painlevé VI

The Laplacian operator applied to $P$ yields

$\displaystyle\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}^{2}P=\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)P\sum_{i\neq
j}\frac{1}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\qquad=P\sum_{\rho}\sum_{i\neq
j}\frac{{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)}{({z}_{\rho}-u_{i})({z}_{\rho}-u_{j})}$
$\displaystyle\qquad=P\sum_{\rho}\sum_{i\neq
j}\left(({z}_{\rho}-(t+1)+u_{i}+u_{j})+\frac{u_{i}(u_{i}-1)(u_{i}-t)}{(u_{i}-u_{j})({z}_{\rho}-u_{i})}-\frac{u_{j}(u_{j}-1)(u_{j}-t)}{(u_{i}-u_{j})({z}_{\rho}-u_{j})}\right).$

This yields

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}^{2}\Phi$
$\displaystyle=\hbar^{2}\bigg{\langle}P\sum_{\rho}\sum_{i\neq
j}\left(({z}_{\rho}-(t+1)+u_{i}+u_{j})+\frac{u_{i}(u_{i}-1)(u_{i}-t)}{(u_{i}-u_{j})({z}_{\rho}-u_{i})}-\frac{u_{j}(u_{j}-1)(u_{j}-t)}{(u_{i}-u_{j})({z}_{\rho}-u_{j})}\right)\bigg{\rangle}.$

Upon rearranging of the terms we obtain the equation

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}^{2}\Phi=-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}-{z}_{\sigma}({z}_{\sigma}-1)({z}_{\sigma}-t)\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi$
$\displaystyle\qquad{}+t(t-1)\hbar\partial_{t}\Phi+\hbar\sum_{\rho}\big{(}(a+b)({z}_{\rho}-1)({z}_{\rho}-t)+c{z}_{\rho}({z}_{\rho}-t)+d{z}_{\rho}({z}_{\rho}-1)\big{)}\partial_{{z}_{\rho}}\Phi$
$\displaystyle\qquad{}-\hbar\sum_{\rho}{z}_{\rho}^{2}\partial_{{z}_{\rho}}\Phi+\hbar\sum_{\rho}{z}_{\rho}\partial_{{z}_{\rho}}\Phi+\big{(}2\hbar
Nm-\hbar m-\hbar^{2}m^{2}\big{)}\sum_{\rho}{z}_{\rho}\Phi$
$\displaystyle\qquad{}+\big{(}\hbar^{2}Nm-\hbar tN^{2}m-\hbar
N^{2}m+m^{2}\hbar^{2}Nt-d\hbar Nmt+\hbar Nmt+(b+d)\hbar Nm\big{)}\Phi$
$\displaystyle\qquad{}+(\hbar
N^{2}-\hbar^{2}N)\bigg{\langle}\sum_{i}u_{i}P\bigg{\rangle}+t(t-1)\hbar
N\bigg{\langle}\sum_{i}\frac{d}{t-u_{i}}P\bigg{\rangle}.$ (A.2)

Now, instead of applying the Euler operator, we consider the operator

${\mathbb{L}}:=\sum_{i}\partial_{u_{i}}(u_{i}(1-u_{i})).$

We apply $\mathbb{L}$ to the integrand

$J(u,z):=\prod_{1\leq i<j\leq
m}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-a-b-1}(1-u_{i})^{-c-1}(t-u_{i})^{-d}.$

After some calculations we obtain

$\displaystyle\mathbb{L}J(\vec{u},\vec{z})=\sum_{i}\bigg{[}(1-2u_{i})\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-a-b-1}(1-u_{i})^{-c-1}(t-u_{i})^{-d}$
$\displaystyle{}+u_{i}(1-u_{i})\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\bigg{(}2\hbar\sum_{i\neq
j}\frac{1}{u_{i}-u_{j}}\bigg{)}\prod_{\rho,i}({z}_{\rho}-u_{i})u_{i}^{-a-b-1}(1-u_{i})^{-c-1}(t-u_{i})^{-d}$
$\displaystyle{}+u_{i}(1-u_{i})\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}({z}_{\rho}-u_{i})\bigg{(}\sum_{\rho}\frac{-1}{{z}_{\rho}-u_{i}}\bigg{)}\prod_{i}u_{i}^{-a-b-1}(1-u_{i})^{-c-1}(t-u_{i})^{-d}$
$\displaystyle{}+u_{i}(1-u_{i})\prod_{i<j}(u_{i}-u_{j})^{2\hbar}\prod_{\rho,i}\frac{({z}_{\rho}-u_{i})}{u_{i}^{a+b+1}(1-u_{i})^{c+1}(t-u_{i})^{d}}$
$\displaystyle\quad{}\times\left(\frac{-a-b-1}{u_{i}}+\frac{c+1}{1-u_{i}}+\frac{d}{t-u_{i}}\right)\bigg{]}$
$\displaystyle{}=\Delta^{2\hbar}\bigg{[}mP-2\sum_{i}u_{i}P-2\hbar(m-1)\sum_{i}u_{i}P-m\sum_{\rho}{z}_{\rho}P-N\sum_{i}u_{i}P+NmP$
$\displaystyle{}-\sum_{\rho}{z}_{\rho}(1-{z}_{\rho})\partial_{{z}_{\rho}}P+(-a-b-d-1)mP+tdmP$
$\displaystyle{}+(a+b+c+d+2)\sum_{i}u_{i}P-t(t-1)\sum_{i}\frac{d}{t-u_{i}}P\bigg{]}\prod_{i}\Theta(u_{i}).$
(A.3)

Integrating (A.3), one obtains zero because the integrand $\mathbb{L}J$ can be
viewed as the divergence of a vector field. Therefore,

$\displaystyle
t(t-1)\bigg{\langle}\sum_{i}\frac{d}{t-u_{i}}P\bigg{\rangle}=(-\hbar
m+Nm+(-b-d)m+tdm)\Phi$
$\displaystyle\hphantom{t(t-1)\bigg{\langle}\sum_{i}\frac{d}{t-u_{i}}P\bigg{\rangle}=}{}+(-2\hbar(m-1)-N+a+b+c+d)\bigg{\langle}\sum_{i}u_{i}P\bigg{\rangle}$
$\displaystyle\hphantom{t(t-1)\bigg{\langle}\sum_{i}\frac{d}{t-u_{i}}P\bigg{\rangle}=}{}-m\sum_{\rho}{z}_{\rho}\Phi-\sum_{\rho}{z}_{\rho}(1-{z}_{\rho})\partial_{{z}_{\rho}}\Phi.$

Substituting the l.h.s. into the equation (A.2) yields the following result

$\displaystyle\hbar^{2}\sum_{\rho}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}^{2}\Phi=-\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}-{z}_{\sigma}({z}_{\sigma}-1)({z}_{\sigma}-t)\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}\Phi$
$\displaystyle\qquad{}+t(t-1)\hbar\partial_{t}\Phi+\hbar\sum_{\rho}\big{(}(a+b)({z}_{\rho}-1)({z}_{\rho}-t)+c{z}_{\rho}({z}_{\rho}-t)+d{z}_{\rho}({z}_{\rho}-1)\big{)}\partial_{{z}_{\rho}}\Phi$
$\displaystyle\qquad{}+\hbar(1-N)\sum_{\rho}{z}_{\rho}(1-{z}_{\rho})\partial_{{z}_{\rho}}\Phi+\hbar
m(N-1-\hbar m)\sum_{\rho}{z}_{\rho}\Phi+\hbar mN(\hbar m+1-N)t\Phi.$

Therefore, the general Hamiltonian operator for quantum Painlevé VI equation
is given by

$\displaystyle t(t-1)H_{\rm
VI}=\hbar\underset{\rho\neq\sigma}{\sum_{\rho,\sigma}}\frac{{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}-{z}_{\sigma}({z}_{\sigma}-1)({z}_{\sigma}-t)\partial_{{z}_{\sigma}}}{{z}_{\rho}-{z}_{\sigma}}$
$\displaystyle\hphantom{t(t-1)H_{\rm
VI}=}{}+\sum_{\rho}\hbar^{2}{z}_{\rho}({z}_{\rho}-1)({z}_{\rho}-t)\partial_{{z}_{\rho}}^{2}$
$\displaystyle\hphantom{t(t-1)H_{\rm
VI}=}{}-\hbar\sum_{\rho}\big{(}(a+b)({z}_{\rho}-1)({z}_{\rho}-t)+c{z}_{\rho}({z}_{\rho}-t)+(d+N-1){z}_{\rho}({z}_{\rho}-1)\big{)}\partial_{{z}_{\rho}}$
$\displaystyle\hphantom{t(t-1)H_{\rm VI}=}{}-\hbar m(N-1-\hbar
m)\sum_{\rho}{z}_{\rho}-\hbar mN(\hbar m+1-N)t.$

######  Remark A.1.

All these operators reduce to the Hamiltonian operators in (3.2) for $N=1$.

### Acknowledgements

The work of F.M. and M.B. was supported in part by the Natural Sciences and
Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06660.

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