## Small Oscillations of a Vortex Ring: Hamiltonian Formalism and Quantization

S.V. TALALOV

Department of Applied Mathematics, Togliatti State University,
14 Belorusskaya str., Tolyatti, Samara region, 445020 Russia.
<EMAIL_ADDRESS>
###### Abstract

This article investigates small oscillations of a vortex ring with zero
thickness that evolves under the Local Induction Equation (LIE). We deduce the
differential equation that describes the dynamics of these oscillations. We
suggest the new approach to the Hamiltonian description of this dynamic
system. This approach is based on the extension of the set of dynamical
variables by adding the circulation $\Gamma$ as a dynamical variable. The
constructed theory is invariant under the transformations of the Galilei
group. The appearance of this group allows for a new viewpoint on the energy
of a vortex filament with zero thickness. We quantize this dynamical system
and calculate the spectrum of the energy and acceptable circulation values.
The physical states of the theory are constructed with help of coherent states
for the Heisenberg -Weyl group.

keywords: vortex ring, constrained Hamiltonian systems, quantization

PACS numbers: 47.10.Df 47.32.C

##  1 Introduction

The study of various vortex structures has a long history. In spite of this
fact, dynamics of such objects continue to attract interest[1]. Without trying
to make a review of the literature on this topic, we will mention only some
works that are directly relevant to our research. So, in the work [2] the
vortex filament (in the LIE approximation) was described firstly in terms of
solutions of non-linear Schrödinger equation. It should also be mentioned that
gauge equivalence between the non-linear Schrödinger equation and the
continuous Heisenberg spin chain exists[3]. It is well-known that quantization
of similar non-linear systems is a complicated problem (see, for example,
[4]). As a consequence, the connection with the initial hydrodynamical system
(the vortex ring in our case) maybe seems not obvious. That is why the
investigations of the small perturbations of certain initial and stable
configurations are interesting, both classical and quantum. Let us note here
the work [5], where, in particular, the system of small-perturbed
straightforward vortex filaments with certain interactions were investigated.
In the work [6], the quantization of such filaments was considered. It is also
necessary to mention the direction of the study of quantum vortices in
superfluid helium [7, 8, 9]. Since our research lies in a different plane, we
will not dwell on this in detail.

In this work, we consider the closed evolving curve
${\boldsymbol{r}}(\tau,\xi)$ that is defined by the formula

${\boldsymbol{r}}(\tau,\xi)=\boldsymbol{q}+{R_{0}}\,\int\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]{\boldsymbol{j}}(\tau,\eta)d\eta\,.$
(1)

Both parameter $\xi$ that parametrizes the curve ${\boldsymbol{r}}(\cdot,\xi)$
and evolution parameter $\tau$ are dimensionless parameters. The constant
$R_{0}$ defines the scale of length. The notation $[\,x\,]$ means the integer
part of the number $x/{2\pi}$, the variables
$\boldsymbol{q}=\boldsymbol{q}(\tau)$ may be $\tau$-depended (conditionally,
these are the coordinates of the ”mass center”). $2\pi$-periodical vector
function ${\boldsymbol{j}}(\xi)\in E_{3}$ defines unit tangent vector. We
postulate that function ${\boldsymbol{r}}(\tau,\xi)$ satisfies the LIE
equation

$\partial_{\tau}{\boldsymbol{r}}(\tau,\xi)=\frac{1}{R_{0}}\,\partial_{\xi}{\boldsymbol{r}}(\tau,\xi)\times\partial_{\xi}^{\,2}{\boldsymbol{r}}(\tau,\xi)\,.$
(2)

Consequently, the function ${\boldsymbol{j}}(\xi)$ satisfies the equation for
the continuous Heisenberg spin chain

$\partial_{\tau}{\boldsymbol{j}}(\tau,\xi)=\,{\boldsymbol{j}}(\tau,\xi)\times\partial_{\xi}^{\,2}{\boldsymbol{j}}(\tau,\xi)\,.$
(3)

The following equalities are fulfilled too:

$\int\limits_{0}^{2\pi}{j}_{k}(\xi)\,d\xi=0\,\qquad(k=x,y,z)\,,$ (4)

The original space-time symmetry group for this system is the group
$E(3)\times E_{\tau}$, where the group $E(3)$ is the group of motions of space
$E_{3}$ and $E_{\tau}$ is the group of ”translations” $\tau\to\tau+c$.

The standard simplest configuration here will be the vortex ring with radius
$R_{0}$ that moves parallel to $z$-axis with some constant velocity:

${\boldsymbol{r}_{0}}(\tau,\xi)=\boldsymbol{q}_{0}+{R_{0}}\,\int\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]{\boldsymbol{j}}_{0}(\tau,\eta)d\eta\,.$
(5)

Tangent vector ${\boldsymbol{j}}_{0}$ has the following coordinates:

${\boldsymbol{j}_{0}}(\xi)=\\{-\sin\xi\,,\quad\cos\xi\,,\quad 0\\}\,.$ (6)

The quantities ${q_{0}}_{i}=const$ for the indexes $i=x,y$ here; the quantity
${q_{0}}_{z}$ is some linear function on the variable $\tau$ that will be
specified latter.

We consider the small perturbation $(\varepsilon<<1)$ of the tangent vector
(6) and coordinates $\boldsymbol{q}_{0}$:

$\boldsymbol{q}=\boldsymbol{q}_{0}+\varepsilon\,\boldsymbol{q}_{prt}\,,\qquad\boldsymbol{j}(\tau,\xi)={\boldsymbol{j}_{0}}(\xi)+\varepsilon{\boldsymbol{j}_{prt}}(\tau,\xi)\,.$
(7)

Therefore, we have representation

${\boldsymbol{r}}(\tau,\xi)={\boldsymbol{r}_{0}}(\tau,\xi)+\varepsilon{\boldsymbol{r}_{prt}}(\tau,\xi)\,.$

Let us substitute the representation (7) for the vector
${\boldsymbol{j}}(\xi)$ into equation (3). Taking into account the equality
$\partial^{\,2}_{\xi}{\boldsymbol{j}_{0}}(\xi)=-{\boldsymbol{j}_{0}}(\xi)$ and
neglecting the terms of order $\varepsilon^{2}$, we deduce the equation

$\partial_{\tau}{\boldsymbol{j}_{prt}}(\tau,\xi)={\boldsymbol{j}_{0}}(\tau,\xi)\times\left[\,{\boldsymbol{j}_{prt}}(\tau,\xi)+\partial_{\xi}^{\,2}{\boldsymbol{j}_{prt}}(\tau,\xi)\,\right]\,.$
(8)

Thus, we have the linear equation that describes the dynamics of small
perturbations for the vortex ring (5). Next, we will not write the
${{}^{\prime\prime}prt^{\prime\prime}}$ index explicitly, hoping that this
will not lead to misunderstandings. It must be emphasized that equalities (4)
must be fulfilled strongly for all configurations (perturbed or not
perturbed).

The symmetry of the initial configuration ${\boldsymbol{j}_{0}}(\tau,\xi)$
makes natural using the cylindrical coordinates $\\{\rho,\phi,z\\}$. The
z-axis of such a system coincides with the axis of the unperturbed vortex
ring. Let three vectors
$\\{{\boldsymbol{e}_{\rho}}\,,{\boldsymbol{e}_{\phi}}\,,{\boldsymbol{e}_{z}}\,\\}$
denote the local basis of the cylindrical system. Obviously, in special case
when $\boldsymbol{j}\equiv{\boldsymbol{j}_{0}}$, the parameter $\xi=\phi$.
Therefore,

${\boldsymbol{j}_{0}}={\boldsymbol{e}_{\phi}}\,.$

Because we consider the small perturbations of the function
${\boldsymbol{j}_{0}}$ only, we assume $\xi=\phi$ for all configurations.

Thus, the equation (8) takes the following form in the cylindrical basis:

$\partial_{\tau}{\boldsymbol{j}}(\tau,\xi)=\Bigl{(}{{j}_{z}}(\tau,\xi)+\partial_{\xi}^{\,2}{{j}_{z}}(\tau,\xi)\Bigr{)}{\boldsymbol{e}_{\rho}}-\Bigl{(}\partial_{\xi}^{\,2}{{j}_{\rho}}(\tau,\xi)-2\,\partial_{\xi}{j}_{\phi}(\tau,\xi)\Bigr{)}\boldsymbol{e}_{z}\,.$
(9)

This equation demonstrates that $\partial_{\tau}{j_{\phi}}(\tau,\xi)\equiv
0\,.$ For example, the initial data ${j_{\phi}}(0,\xi)\equiv j_{\phi}^{\,0}$,
where $\partial_{\xi}j_{\phi}^{\,0}\equiv 0$, lead to the equality

${j_{\phi}}(\tau,\xi)\equiv j_{\phi}^{\,0}\,,\qquad j_{\phi}^{\,0}=const\,.$

Obviously, the perturbation
${\boldsymbol{j}_{0}}(\xi)\to{\boldsymbol{j}_{0}}(\xi)+\varepsilon
j_{\phi}^{\,0}{\boldsymbol{e}_{\phi}}$ conserves the form of the original
vortex ${\boldsymbol{r}_{0}}$. In our subsequent considerations we will
consider the case $j_{\phi}^{\,0}=const$ only. Therefore, the perturbation
amplitude ${\boldsymbol{j}}(\tau,\xi)$ of the vector
${\boldsymbol{j}_{0}}(\tau,\xi)$ has following form in the cylindrical basis:

${\boldsymbol{j}}(\tau,\xi)=j_{\rho}(\tau,\xi){\boldsymbol{e}}_{\rho}+j_{\phi}^{\,0}\boldsymbol{e}_{\phi}+j_{z}(\tau,\xi){\boldsymbol{e}_{z}}\,.$
(10)

The following identity takes place:

${\boldsymbol{j}}(\tau,\xi){\boldsymbol{j}_{0}}(\tau,\xi)\equiv
j_{\phi}^{\,0}\,.$ (11)

The notation ${\boldsymbol{j}}{\boldsymbol{j}_{0}}$ means an inner product of
two vectors. In the future, it is more convenient for us to consider the non-
trivial components of the vector ${\boldsymbol{j}}$ in a complex-valued form.
We introduce the notation ${\sf j}$ for the complex perturbation amplitude to
avoid any ambiguity. Thus,

${\sf j}=j_{\rho}+ij_{z}\,.$

Equation (9) writing in complex form as follows:

$\partial_{\tau}{\sf j}=-i\partial_{\xi}^{\,2}{\sf j}-\frac{i}{2}\Bigl{(}{\sf
j}-\overline{\,\sf j\,}\,\Bigr{)}\,.$ (12)

This simple equation can be solved explicitly:

${\sf j}(\tau,\xi)=\sum_{n}{\sf
j}_{\,n}\,e^{\,i\,[n\xi+n\sqrt{n^{2}-1}\,\tau\,]}\,,$ (13)

where ${\sf j}_{\,n}\equiv const$ and the coefficients $\overline{\,{\sf
j}\,}_{\,-n}$ and ${\sf j}_{\,n}$ are related to each other as follows:

$\overline{\,{\sf
j}\,}_{\,-n}=2\left[n\sqrt{n^{2}-1}-n^{2}+\frac{1}{2}\right]{\sf j}_{\,n}\,.$
(14)

It is clear that this solution is stable for $\tau\to\infty$. Note that the
stability of the real vortex rings is a non-trivial problem in general. For
example, this problem was investigated in the work [10] for the Norbury rings.

Restrictions (4) are rewriting in terms of the cylindrical coordinates as
follow:

$\int_{0}^{2\pi}j_{\rho}(\xi)\,e^{\pm
i\xi}d\xi=0\,,\qquad\int_{0}^{2\pi}j_{z}(\xi)\,d\xi=0\,.$ (15)

As regards to the coefficients ${\sf j}_{\,0}$, ${\sf j}_{\pm 1}$ in the
solution (13), constraints (15) lead to the formulas

$\displaystyle{\sf j}_{\,0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\Bigl{(}j_{\rho}+ij_{z}\Bigr{)}d\xi=\frac{1}{2\pi}\int_{0}^{2\pi}j_{\rho}\,d\xi\,,$
(16) $\displaystyle{\sf j}_{\,\pm 1}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\Bigl{(}j_{\rho}+ij_{z}\Bigr{)}\,e^{\mp
i\xi}d\xi=\frac{1}{2\pi}\int_{0}^{2\pi}j_{z}(\pm\sin\xi+i\cos\xi)\,d\xi\,.$
(17)

So, last equalities lead to the restrictions for the coefficients ${\sf
j}_{\pm 1}$:

${\sf j}_{\,1}=-\overline{\,{\sf j}\,}_{\,-1}\,.$ (18)

As regards the coefficient ${\sf j}_{\,0}$, the equalities (16) lead to the
restriction ${\sf j}_{\,0}=\,\overline{\,{\sf j}\,}_{\,0}$. Although these
restrictions are deduced from the constraint (4), they are consistent with the
formula (14).

##  2 Dynamical invariants and extended set of the variables

The consideration of vortex structure in terms of the equation (2) only is
apparently too formal. As an addition to the LIE equation, we postulate in our
theory the canonical formulas for the momentum and angular momenta, that was
deduced in the fluid dynamics [11]:

$\tilde{\boldsymbol{p}}=\frac{1}{2}\,\int\,\boldsymbol{r}\times\boldsymbol{\omega}(\boldsymbol{r})dV\,,\qquad\tilde{\boldsymbol{s}}=\frac{1}{3}\,\int\,\boldsymbol{r}\times\bigl{(}\boldsymbol{r}\times\boldsymbol{\omega}(\boldsymbol{r})\bigr{)}dV\,.$
(19)

The vector $\boldsymbol{\omega}(\boldsymbol{r})$ means the vorticity. The
fluid density $\varrho\equiv 1$ here.

As well-known, the vorticity of the closed vortex filament calculates by means
of the formula

$\boldsymbol{\omega}(\boldsymbol{r})=\Gamma\int\limits_{0}^{2\pi}\,\hat{\delta}(\boldsymbol{r}-\boldsymbol{r}(\xi))\partial_{\xi}{\boldsymbol{r}}(\xi)d\xi\,,$
(20)

where the symbol $\Gamma$ denotes the circulation and the symbol
$\hat{\delta}(\xi)$ means $2\pi$-periodical $3D$ $\delta$-function.

Taking into account the formulae (1), (4) and (20), the following expression
for the canonical momentum is deduced:

$\tilde{\boldsymbol{p}}={R_{0}}^{2}\Gamma{\boldsymbol{f}}\,,\qquad{\boldsymbol{f}}=\frac{1}{2}\iint\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]\,{\boldsymbol{j}}(\eta)\times{\boldsymbol{j}}(\xi)d\xi
d\eta\,.$ (21)

A similar formula can be written for the angular momenta
$\tilde{\boldsymbol{s}}$; however, we omit the relevant details here. As
opposed to formulae for the values $\tilde{\boldsymbol{p}}$ and
$\tilde{\boldsymbol{s}}$, the canonical formula for the energy ${\mathcal{E}}$
gives the unsatisfactory result because of the divergence of the integral. We
will return to this question later.

Let us substitute the expansion (7) into expression for the vector
${\boldsymbol{f}}$ (see (21)). Taking into account identity (11), the
following formula for the vector ${\boldsymbol{f}}$ holds:

${\boldsymbol{f}}=\pi(1+2\,\varepsilon
j_{\phi}^{\,0}){\boldsymbol{e}}_{z}-\varepsilon{\boldsymbol{f}}_{\\!\\!\perp}\,,\qquad{\boldsymbol{f}}_{\\!\\!\perp}=\int_{0}^{2\pi}j_{z}(\xi){\boldsymbol{e}}_{\phi}\,d\xi\,.$
(22)

The quantity ${\boldsymbol{f}}$ includes both unperturbed
($\pi{\boldsymbol{e}}_{z}$) and perturbed ($2\pi\,\varepsilon
j_{\phi}^{\,0}{\boldsymbol{e}}_{z}-\varepsilon{\boldsymbol{f}}_{\\!\\!\perp}$)
values. Correspondingly, the full momentum $\tilde{\boldsymbol{p}}$ has
following form:

$\tilde{\boldsymbol{p}}=-\varepsilon\tilde{\boldsymbol{p}}_{\\!\perp}+\tilde{p}_{\,\parallel}{\boldsymbol{e}_{z}}\,,$

where

$\tilde{p}_{\,\|}=\pi(1+2\,\varepsilon j_{\phi}^{\,0}){R_{0}}^{2}\Gamma\,.$
(23)

As for the amplitudes of perturbations for the momentum, we have the following
formulas:

$\displaystyle\tilde{p}_{z}$ $\displaystyle=$ $\displaystyle
2\pi{R_{0}}^{2}\Gamma j_{\phi}^{\,0}\,$ (24)
$\displaystyle\tilde{\boldsymbol{p}}_{\\!\perp}$ $\displaystyle=$
$\displaystyle{R_{0}}^{2}\Gamma{\boldsymbol{f}}_{\\!\\!\perp}\,.$ (25)

Next, we intend to construct a Hamiltonian dynamical system that corresponds
to the equation (8). For the general case (2), the corresponding approach was
proposed by the author in the work [12]. We will briefly recall here the main
points of the proposed theory, making the necessary modifications along the
way.

In our opinion, the following steps must be done to construct the physically
interpreted dynamical system in our case:

  1. 1.

As mentioned above, we need to supplement the equation (2) with formulas for
the momentum $\tilde{\boldsymbol{p}}$ and the angular momenta
$\tilde{\boldsymbol{s}}$;

  2. 2.

The variable ”velocity of liquid” is absent in our theory. We propose to take
into account the dynamics of the surrounding fluid in a minimal way: to
declare the value $\Gamma$ as a dynamic variable, in addition to variables
$\boldsymbol{j}(\xi)$ and $\boldsymbol{q}$. We denote as ${\mathcal{A}}$ this
(extended) set of the dynamical variables
$\\{\,\boldsymbol{q}\,,{\boldsymbol{j}}(\xi)\,,\Gamma\,\\}$ constrained by the
conditions (4).

  3. 3.

The theory must contain a sufficient number of dimensional constants. So far,
we use the dimensionless ”time” $\tau$. Consequently, the additional
dimensional constant $t_{0}$ that defines the scale of time, must be added to
the theory in some way. Subsequently, we will express this constant in terms
of other dimensional constants that have a clear physical meaning in our
model. In addition to the $R_{0}$ and $t_{0}$ constants, the theory should
contain a ”mass constant” $m_{0}$. This constant will appear in the theory in
a completely natural way later on.

As a subtotal, we have the following

###### Proposition 1

The set ${\mathcal{A}}$ parametrizes the considered dynamical system - the
closed vortex filament ${\boldsymbol{r}}(\xi)$ that evolving in accordance
with the LIE equation. This dynamical system has a momentum
$\tilde{\boldsymbol{p}}$ and angular momenta $\tilde{\boldsymbol{s}}$ that
calculated as prescribed above.

To perform the hamiltonization of our system, we are going to describe the set
${\mathcal{A}}$ in terms of other variables. The reasons are following:

  * •

we intend to expand the symmetry group $E(3)\times E_{\tau}$ to Galilei group
${\mathcal{G}}_{3}$ and use the group-theoretical approach for the definition
of the energy of our system;

  * •

new variables will be more suitable for subsequent quantization that will be
fulfilled later in this article.

In addition, the new variables will give the obvious interpretation of the
considered dynamical system as some structured particle. It is probably
appropriate to mention here the pioneering work [13] in which the observed
particles are modeled by vortex structures.

As a first step, we extend the set ${\mathcal{A}}$. Let us denote as
${\mathcal{A}}^{\,\prime}$ the set of the independent variables
$({\boldsymbol{q}}\,;\,\tilde{\boldsymbol{p}}\,;{\boldsymbol{j}}(\xi)\,)$. The
formula (21) makes the injection $F$:

$F:\quad{\mathcal{A}}\quad\mathrel{\mathop{\longrightarrow}^{F}}\quad{\mathcal{A}}^{\,\prime}\,,\qquad{\rm
Ran}\,F\subset{\mathcal{A}}^{\,\prime}\,.$

On the set ${\mathcal{A}}^{\,\prime}$ the action of the central extended
Galilei group $\widetilde{\mathcal{G}}_{3}$ is defined by natural way. We
parametrize the elements $g\in\widetilde{\mathcal{G}}_{3}$ as follows:

$g:\qquad\Bigl{(}{\mathcal{R}}\,,{\boldsymbol{v}}\,,{\boldsymbol{a}}\,,c\,;m_{0}\Bigr{)}\,$

where ${\mathcal{R}}\in SO(3)$, ${\boldsymbol{v}}\,,{\boldsymbol{a}}\in
E_{3}$, $c\in{\sf R}$ and the central charge $m_{0}\in{\sf R}$. Traditionally,
the last parameter is interpreted as ”mass of the particle”. Before
determining the action of this group on the set ${\mathcal{A}}^{\,\prime}$, we
introduce the factor $m_{0}/R_{0}^{3}$ in standard hydrodynamical formulas
(19) to provide the dimension for the values $\tilde{\boldsymbol{p}}$ and
$\tilde{\boldsymbol{s}}$ as in classical mechanics:

$\tilde{\boldsymbol{p}}\longrightarrow{\boldsymbol{p}}=(m_{0}/R_{0}^{3})\tilde{\boldsymbol{p}}\,,\qquad\tilde{\boldsymbol{s}}\longrightarrow{\boldsymbol{s}}=(m_{0}/R_{0}^{3})\tilde{\boldsymbol{s}}\,.$
(26)

Taking into account this redefinition, the group action $\circ$ on the set
${\mathcal{A}}^{\,\prime}$ is defined as follows:

$g\circ({\boldsymbol{q}}\,;\,{\boldsymbol{p}}\,;{\boldsymbol{j}}(\xi)\,)=({\mathcal{R}}{\boldsymbol{q}}+{\boldsymbol{v}}t_{0}\tau+{\boldsymbol{a}}\,;\,{\mathcal{R}}{\boldsymbol{p}}+m_{0}{\boldsymbol{v}}\,;{\mathcal{R}}{\boldsymbol{j}}(\xi)\,)\,,$

and $g\circ(t_{0}\tau)=t_{0}\tau+c$.

Next, we introduce the variables

$q_{i}(0)=q_{i}-{\tau}(t_{0}/m_{0})p_{i}\,,\qquad i=x,y,z\,,$

that will be convenient sometimes for using. The curve
${\boldsymbol{r}}(\tau,\xi)$ is reconstructed through variables
$({\boldsymbol{q}}(0)\,;\,{\boldsymbol{p}}\,;{\boldsymbol{j}}(\xi)\,)$ in
accordance with the formula

${\boldsymbol{r}}(\tau,\xi)={\boldsymbol{q}(0)}+\tau(t_{0}/m_{0}){\boldsymbol{p}}+{R_{0}}\int\limits_{0}^{2\pi}\left[\,{\xi-\eta}\,\right]{\boldsymbol{j}}(\tau,\eta)d\eta\,.$
(27)

As a second step, we must introduce certain constraints on the set
${\mathcal{A}}^{\,\prime}$ that define set
$\Omega\subset{\mathcal{A}}^{\,\prime}$. Criteria for introducing these
constraints - the one-to-one correspondence

${\mathcal{A}}\longleftrightarrow\Omega\,.$

It is clear that we must define two constraints, because we introduce three
variables $({\boldsymbol{p}})$ instead one variable $\Gamma$.

Fist of all, we must require that the vectors ${\boldsymbol{p}}_{\\!\perp}$
and ${\boldsymbol{f}}_{\\!\\!\perp}$ are proportional. In general, it is not
true on set ${\mathcal{A}}^{\,\prime}$, because these vectors are independent
at this set. For convenience, let us introduce the complex values:

${\sf p}=(p_{\\!\perp})_{x}+i(p_{\\!\perp})_{y}\,,\qquad{\sf
f}=({f}_{\\!\perp})_{\,x}+i({f}_{\\!\perp})_{\,y}\,.$

In accordance both the definition of the vector ${\boldsymbol{f}}_{\\!\perp}$
(see (22)) and the formulas (17) we have the equalities:

${\sf f}=-\int_{0}^{2\pi}\\!j_{z}\sin\xi d\xi+i\int_{0}^{2\pi}\\!j_{z}\cos\xi
d\xi=2\pi{\sf j}_{-1}\,.$ (28)

Taking into account the replacement (26), formula (25) takes following form:

${\sf p}=\frac{m_{0}\Gamma}{R_{0}}\,{\sf f}=\frac{2\pi
m_{0}\Gamma}{R_{0}}\,{\sf j}_{-1}\,.$ (29)

Therefore, by virtue of the formula (25) and the complex-valued notations for
the corresponding values, we must demand:

$\exists\lambda\in{\sf R}:\qquad\quad{\sf p}=2\pi\lambda p_{\,0}{\sf
j}_{-1}\,.$ (30)

The variables ${\sf p}$ and ${\sf j}_{-1}$ are the complex numbers here and
the value $p_{\,0}=m_{0}R_{0}/t_{0}$. As we will show further, the constant
$p_{\,0}$ has a clear physical meaning, so it seems natural to use it as in-
put constant, instead of a value $t_{0}$. However, we will use the constant
$t_{0}$ to simplify some formulas.

The condition (30) can also be written in following form:

$p_{x}(f_{\\!\perp})_{y}-p_{y}(f_{\\!\perp})_{x}=0\,.$ (31)

Of couse, this equality (just like equality (30)) is fullfilled identically in
the case when our theory is parametrized by the set ${\mathcal{A}}$. In
complex-valued notations the constraint (31) is written as follows:

$\Phi_{0}\equiv{\sf p}\,{\overline{\,\sf j\,}_{-1}}-{\overline{\sf p}}\,{{\sf
j}_{-1}}=0\,.$ (32)

If the quantities ${\sf p}$ and ${\sf j}_{-1}$ take non-zero values and the
condition (32) is fulfilled, the variable $\Gamma$ can be determined
unambiguously through the formulas:

$\Gamma=\frac{\lambda R_{0}^{2}}{t_{0}}\,,\qquad\quad|{\sf
p}|^{2}-4\pi^{2}p_{0}^{\,2}\lambda^{2}|{\sf j}_{-1}|^{2}=0\,.$ (33)

In corresponding zero points the value $\Gamma=\Gamma_{0}$ is still
undetermined. Let us substitute the representation (27) for the original
vortex ${\boldsymbol{r}_{0}}(\tau,\xi)$ in LIE equation (2). This procedure
leads to the equality for the momentum unperturbed vortex ring:

$p_{z}=p_{\,0}=m_{0}R_{0}/t_{0}\,.$

To deduce this formula we suppose that
$\partial_{\tau}{\boldsymbol{p}}=\partial_{\tau}{\boldsymbol{q}}(0)=0$. These
conditions will be coordinated with the subsequent hamiltonization of our
dynamical system. In accordance with formula (23) and (26) we have for this
case: $p_{\,0}=\pi m_{0}\Gamma_{0}/{R_{0}}\,.$ Therefore,
$\Gamma_{0}=R_{0}^{2}/\pi t_{0}$.

Let us return to the perturbed case. In this case the value $\Gamma$ in the
formula (24) must be same value as was determined in formula (33). That is why
we must write the second constraint:

$\Phi_{1}\equiv|{\sf j}_{-1}|^{2}p_{z}^{2}-(j_{\phi}^{\,0})^{2}|{\sf
p}|^{2}=0\,.$ (34)

In special case when the value $j_{\phi}^{\,0}=0$, we have the constraint

$\Phi_{1}\equiv p_{z}=0\,$ (35)

instead constraint (34). This means that set $\Omega$ describes the planar
system here. We will consider the case $j_{\phi}^{\,0}=0$ only. In a general
case when $j_{\phi}^{\,0}\not=0$, the set
$\Omega\subset{\mathcal{A}}^{\,\prime}$ defines by constraints (32) and (34).

Finally, we have the following

###### Proposition 2

The variables ${\boldsymbol{j}}(\xi)$, $\boldsymbol{q}$, $\boldsymbol{p}$,
that are declared as the new fundamental variables, parametrize uniquely
considered dynamical system. These variables are constrained by the equalities
(32) and (34).

##  3 Energy and Hamiltonian structure

The straightforward calculation of the energy of a vortex filament is usually
performed using the canonical formula [14]

${\mathcal{E}}=\frac{1}{8\pi}\,\iint\frac{\boldsymbol{\omega}(\boldsymbol{r})\boldsymbol{\omega}(\boldsymbol{r}^{\prime})}{|\,\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\,dVdV^{\prime}=\frac{{\Gamma}^{\,2}}{8\pi}\iint\frac{\partial_{\xi}{\boldsymbol{r}}(\xi)\partial_{\xi}{\boldsymbol{r}}(\xi^{\prime})}{|\,{\boldsymbol{r}}(\xi)-{\boldsymbol{r}}(\xi^{\prime})|}\,d\xi
d\xi^{\prime}\,,$

The result is unsatisfactory if the filament has zero thickness: the integral
in this formula diverges. The standard approach to solve this problem is to
take into account the finite thickness $a$ of the filament and the subsequent
regularization of the integral (see, for example, [15], where the interaction
between pairs of quantized vortex rings was studied).

In the proposed approach, we have chosen a different method: the energy of the
arbitrary configuration in our model will be considered from the group-
theoretical viewpoint. Indeed, the Lee algebra of group
$\widetilde{\mathcal{G}}_{3}$ has three Cazimir functions:

${\hat{C}}_{1}=m_{0}{\hat{I}}\,,\quad{\hat{C}}_{2}=\left({\hat{M}}_{i}-\sum_{k,j=x,y,z}\epsilon_{ijk}{\hat{P}}_{j}{\hat{B}}_{k}\right)^{2}\quad{\hat{C}}_{3}=\hat{H}-\frac{1}{2m_{0}}\sum_{i=x,y,z}{\hat{P}}_{i}^{\,2}\,,$

where ${\hat{I}}$ is the unit operator, ${\hat{M}}_{i}$, $\hat{H}$,
${\hat{P}}_{i}$ and ${\hat{B}}_{i}$ ($i=x,y,z$) are the respective generators
of rotations, time and space translations and Galilean boosts. As it is well
known, the function ${\hat{C}}_{3}$ can be interpreted as an ”internal energy
of the particle”. Because our dynamical system has an ”internal degrees of the
freedom”, the function ${\hat{C}}_{3}$ can depend on the internal variables.
We define these functions as follows

${C}_{3}={\mathcal{E}}_{0}\sum_{n>1}|\,{\sf j}_{\,-n}|^{2}n\sqrt{n^{2}-1}\,.$

Here we have introduced into consideration the value
${\mathcal{E}}_{0}=m_{0}R_{0}^{2}/t_{0}^{2}$ which defines the energy scale in
our theory. The choice of the function $C_{3}$ will be quite justified after
the definition of the Hamiltonian structure.

As a result, the following function on the set $\Omega$ is a good candidate
for the energy111We are considering here the energy of excitations only.:

${H}_{0}(p_{1},p_{2},p_{3}\,;{\sf
j})=\frac{{\boldsymbol{p}}^{\,2}}{2m_{0}}+{\mathcal{E}}_{0}\sum_{n>1}|\,{\sf
j}_{\,-n}|^{2}n\sqrt{n^{2}-1}\,.$ (36)

To complete the consideration of energy, we must define the Poisson brackets
that are compatible with the dynamics and constraints.

In this article we consider the simplest case that corresponds the value
$j_{\phi}^{\,0}=0$. Consequently, the constraint (35) is fulfilled. It is
quite natural to add the additional constraint

$\Phi_{2}\equiv q_{z}-R_{0}\tau=0\,.$ (37)

Pursuant to Dirac’s prescriptions about the primacy of Hamiltonian structure,
we define such structure axiomatically here. The correspondent definitions are
following.

  * •

Phase space ${\mathcal{H}}={\mathcal{H}}_{3}\times{\mathcal{H}}_{j}$. The
space ${\mathcal{H}}_{3}$ is the phase space of a $3D$ free structureless
particle. It is parametrized by the variables ${\boldsymbol{q}}$ and
${\boldsymbol{p}}$. The space ${\mathcal{H}}_{j}$ is parametrized by the
quantities ${\sf j}_{\,-n}$ ($n=0,1,\dots$).

  * •

Poisson structure:

$\displaystyle\\{p_{i}\,,q_{j}\\}$ $\displaystyle=$
$\displaystyle\delta_{ij}\,,\qquad i,j=x,y,z\,,$ $\displaystyle\\{{\sf
j}_{\,m},\overline{\,\sf j\,}_{\,n}\\}$ $\displaystyle=$
$\displaystyle(i/{\mathcal{E}}_{0}t_{0})\,\delta_{mn}\,,\qquad
m,n=-1,-2,\dots$ (38)

All other brackets vanish. The variable ${\sf j}_{\,0}$ annulates all
brackets. Thus, the Poisson structure of the theory is degenerate in general:
the value ${\sf j}_{\,0}$ marks the symplectic sheets where the structure will
be non-degenerate.

  * •

Constraints (32), (35) and (37). It is clear that constraints (35) and (37)
form the pair of second type constraints in Dirac terminology. Moreover the
following equalities hold:

$\\{\Phi_{0},\Phi_{k}\\}=0\,\qquad k=1,2.$

Therefore, we can exclude the coordinates $p_{z}$ and $q_{z}$ from the phase
space ${\mathcal{H}}_{3}$ replacing it to the phase space ${\mathcal{H}}_{2}$.
The last one - the phase space of a free structureless particle on a plane.
There are no additional constraints here because the equality
$\\{H,\Phi_{0}\\}=0\,.$

  * •

Hamiltonian

$H=H_{0}+\ell\Phi_{0}\,,$ (39)

where the function $H_{0}$ is defined by the formula (36) with replacing
$\boldsymbol{p}\to\boldsymbol{p}_{\perp}$. The quantity ${\ell}$ is the
Lagrange factor.

Let’s pay attention to the following point. We introduce the set of the new
variables ${\mathcal{A}}^{\,\prime}$ which is more extensive than the set of
original variables. Constraints on set ${\mathcal{A}}^{\,\prime}$ were
postulated. These constraints lead to a certain arbitrariness in dynamics.
That is why the constructed dynamical system is not equivalent to the original
one, but it is more extensive. Indeed, let us define the physical
(dimensional) time $t=t_{0}\tau$. The following Proposition is true:

###### Proposition 3

The following Hamilton equations are valid:

$\displaystyle\frac{\partial\boldsymbol{q}}{\partial t}$ $\displaystyle=$
$\displaystyle\\{H_{0},{\boldsymbol{q}}\\}=\frac{\boldsymbol{p}}{m_{0}}\,,\qquad\frac{\partial\boldsymbol{p}}{\partial
t}=\\{H_{0},\boldsymbol{p}\\}=0\,,$ (40) $\displaystyle\frac{\partial{\sf
j}(\tau,\xi)}{\partial t}$ $\displaystyle=$ $\displaystyle\\{H_{0},{\sf
j}(\tau,\xi)\\}=\frac{i}{t_{0}}\sum_{|n|>1}{\sf
j}_{\,n}n\sqrt{n^{2}-1}\,e^{\,i[\,n\xi+n\sqrt{n^{2}-1}\,\tau\,]}\,.$ (41)

Because the Hamiltonian $H$ differs from $H_{0}$, constructed system is
equivalent to original if the Lagrange factor ${\ell}=0$.

##  4 Quantization

Numerous articles devoted to the problem of turbulence show that the
understanding of this phenomenon is still not full. This statement also fully
applies to the turbulence of quantum liquids. Without setting out to review
the literature on this issue, we note that a number of authors (see, for
example, [16] ) assume that the key to understanding this problem is the
topological defects of such liquids - vortices. That is why the quantum
description of vortices is an important task. The information about the
spectrum of energy of such defects for concrete tasks allows investigating
certain statistical characteristics of the system. In this paper, we aim to
develop a new approach to the quantization of a single closed vortex with zero
thickness. The author believes that in the future, the results obtained may
provide new opportunities for explaining the behavior of quantum liquids.

The constructed Hamiltonian structure opens up possibilities for quantization
of the small perturbations of the vortex ring under study. Firstly, we must
define a Hilbert space $\boldsymbol{H}$ of the quantum states of our dynamical
system. The structure of the phase space ${\mathcal{H}}$ lead to the following
structure:

$\boldsymbol{H}=\boldsymbol{H}_{2}\otimes\boldsymbol{H}_{F}\,,$ (42)

where the symbol $\boldsymbol{H}_{2}$ denotes the Hilbert space of a free
structureless particle on a plane (the space $L^{2}({\sf R}_{2})$ for example)
and symbol $\boldsymbol{H}_{F}$ denotes the Fock space for the infinite number
of the harmonic oscillators. The creation and annihilation operators which are
defined in the space $\boldsymbol{H}_{F}$, have standard commutation relations

$[\,\hat{a}_{m},\hat{a}_{n}^{+}]=\hat{I}_{F}\,,\qquad\hat{a}_{m}|\,0\rangle=0\,,\qquad
m,n=1,2,\dots\,,\qquad|\,0\rangle\in\boldsymbol{H}_{F}\,,$

where the operator $\hat{I}_{F}$ is unit operator in the space
$\boldsymbol{H}_{F}$. Let us quantize our theory. We must to construct the
function $A\to\hat{A}$, where $A$ denotes some classical variable and
$\hat{A}$ denote some operator in the space $\boldsymbol{H}$. Traditionally,
we must to demand

$[\hat{A},\hat{B}]=-i\hbar\widehat{\\{A,B\\}}\,$

if the quantities $A$, $B$, $\dots$ denote the funamental variables in our
theory. This equality can possess of some ”anomalous terms” if the
”observables” $A$, $B$ are the functions of the fundamental variables. These
terms depend on a rule of the ordering of non-commuting operators. We will not
discuss these issues here [17].

Let us consider the case when $\boldsymbol{H}_{2}=L^{2}({\sf R}_{2})$. This
case corresponds to the perturbation of a vortex ring in unbounded space. Our
postulate of quantization is following:

$q_{x,y}\to q_{x,y}\otimes\,\hat{I}_{F}\,,\qquad
p_{x,y}\to-i\hbar\frac{\partial}{\partial
q_{x,y}}\otimes\,\hat{I}_{F}\,,\qquad{\sf
j}_{\,-n}\to\sqrt{\frac{\hbar}{t_{0}{\mathcal{E}}_{0}}}\,(\hat{I}_{2}\otimes\,\hat{a}_{n})\,,$

where $n=1,2,\dots\,$ and operator $\hat{I}_{2}$ is unit operator in the space
$\boldsymbol{H}_{2}$. Next, we will not write the index $n=1$ explicitly:
$\hat{a}_{1}=\hat{a}$ and so on. This simplification will be justified later.
Moreover we will not write the constructions $(\dots\otimes\,\hat{I}_{F})$ and
$(\hat{I}_{2}\otimes\,\dots)$ explicitly, hoping that this will not lead to
misunderstandings.

As a next step, we should to construct the physical subspace
$\boldsymbol{H}_{phys}\subset\boldsymbol{H}$. In accordance with Dirac’s
prescription, the presence of constraint (32) leads to the equation for the
vectors $|\psi\rangle\in\boldsymbol{H}_{phys}$:

${\widehat{\Phi}_{0}}|\psi\rangle=0\,.$

Here it is extremely important to pay attention to the following fact. In a
classical theory, all forms of the first-type constraints lead to the same
theory: in our case, we can assume $\Phi_{0}=0$ or $\Phi_{0}^{\,2}=0$ and so
on. This is not the case at all in the quantum version of the model. The
different forms of first type constraints correspond to the different
equations for the ”physical vectors” in a quantum theory222Apparently, Dirac’s
words are still relevant: ”…methods of quantization are all of the nature of
practical rules, whose application depends on consideration of simplicity”.
For instant, it is clear that the solutions of the equation
${\widehat{\Phi}_{0}}|\psi\rangle=0$ differ from the solutions of the equation
${\widehat{\Phi}_{0}^{\,2}}|\psi\rangle=0$. Consequently, we need to
supplement the quantization rules with a specific choice of the form of the
constraint in classical theory.

Let us investigate this problem in our model in more detail. First of all, we
consider the classical constraint in form (30). Taking into account formula
(28), we search the vectors $|\psi_{phys}\rangle$ so that

$\exists\lambda\in{\sf R}:\qquad(\hat{\sf p}-2\pi\lambda
p_{\,0}\sqrt{\frac{\hbar}{t_{0}{\mathcal{E}}_{0}}}\,\hat{a})|\psi_{phys}\rangle=0\,.$
(43)

Let the complex number ${\sf p}=p_{x}+ip_{y}$ is eigenvalue that corresponds
to the (generalized) eigenvector $|{\sf
p}\rangle\in\boldsymbol{H}_{2}^{\,\prime}$ of the operator $\hat{{\sf p}}$.
The notation $\boldsymbol{H}_{2}^{\,\prime}$ means that prosedure of a rigging
of the space $\boldsymbol{H}_{2}$ must be fulfilled to consider the
generalized eigenvectors rigorously [19]. As ansatz for the solutions of the
equation (43), we use the following form for the ”physical vectors”
$|\psi_{phys}\rangle$:

$|\psi_{phys}({\sf p})\rangle=|{\sf p}\rangle|\psi_{p}\rangle\,,\qquad|{\sf
p}\rangle\in\boldsymbol{H}_{2}^{\,\prime}\,,\quad|\psi_{p}\rangle\in\boldsymbol{H}_{F}\,.$
(44)

Therefore, the equation for the vector $|\psi_{p}\rangle$ takes following
form:

$\exists\lambda\in{\sf R}:\qquad({\sf p}-2\pi\lambda
p_{\,0}\sqrt{\frac{\hbar}{t_{0}{\mathcal{E}}_{0}}}\,\hat{a})|\psi_{p}\rangle=0\,,\quad|\psi_{p}\rangle\in\boldsymbol{H}_{F}\,,\quad{\sf
p}\in{\sf C}\,.$ (45)

In other words, the vectors $|\psi_{p}\rangle$ are the eigenvectors of the
spectral problem (45).

###### Proposition 4

Let the the vectors $|{\sf p}/\lambda\rangle\in\boldsymbol{H}_{F}$ form the
system of coherent states333See, for example, [18] for Heisenberg - Weyl group
with algebra

$[\,\hat{a},\hat{a}^{+}]=\hat{I}\,,\qquad[\,\hat{a}^{+},\hat{I}]=[\,\hat{a},\hat{I}]=0\,$

as follows:

$|{\sf p}/\lambda\rangle=\exp\Bigl{[}({\sf p}\,\hat{a}^{+}-{\overline{{\sf
p}}}\,\hat{a})/2\pi\lambda\sqrt{\frac{\hbar
p_{\,0}}{R_{\,0}}}\,\Bigr{]}\,|0\rangle\,,\qquad\lambda\in{\sf R}\,,\quad{\sf
p}\in{\sf C}\,.$ (46)

Then the equation (45) has solutions

$|\psi^{\lambda}\rangle=|n_{1},\dots,n_{k};{\sf
p}/\lambda\rangle\equiv\hat{a}^{+}_{n_{1}}\hat{a}^{+}_{n_{2}}\dots\hat{a}^{+}_{n_{k}}|{\sf
p}/\lambda\rangle\,,\quad\qquad n_{j}>1\,.$

This statement can be verified directly because the coherent states $|{\sf
p}/\lambda\rangle$ are eigenvectors of the operator $\hat{a}$ with eigenvalue
${\sf p}/(2\pi\lambda\sqrt{{\hbar p_{\,0}}/{R_{\,0}}})$. Recall that the
parameter $\lambda$ has the meaning of ”dimensionless circulation” in our
model.

Let us consider the vectors $|\psi_{[n]}^{\lambda}({\sf
p})\rangle\in\boldsymbol{H}^{\,\prime}$:

$|\psi_{[n]}^{\lambda}({\sf p})\rangle=C_{[n]}|{\sf
p}\rangle|n_{1},\dots,n_{k};{\sf p}/\lambda\rangle\,,\qquad\lambda\in{\sf
R}\,,\quad[n]=n_{1},\dots,n_{k}\,,$ (47)

where numbers $C_{[n]}$ are normalizing coeffisients, so that normalizing
conditions are fulfilled:

$\langle\psi_{[n]}^{\lambda}({\sf p})|\psi_{[m]}^{\lambda}({\sf
p^{\,\prime}})\rangle=\delta_{kl}\delta_{n_{1}m_{1}}\dots\delta_{n_{k}m_{k}}\delta(p_{x}-p_{x}^{\,\prime})\delta(p_{y}-p_{y}^{\,\prime})\,.$

Here we should note that the entire set of vectors (47) can’t form a physical
subspace $\boldsymbol{H}_{phys}\subset\boldsymbol{H}$ because any
superpositions

$c_{1}|\psi_{[n]}^{\lambda_{1}}({\sf
p}_{1})\rangle+c_{2}|\psi_{[n]}^{\lambda_{2}}({\sf
p}_{2})\rangle\,,\qquad\lambda_{1}\not=\lambda_{2}\,,\quad{\sf p}_{1}={\sf
p}_{2}$

is not a solution to the spectral problem (45). Moreover we can’t postulate
”superselection rules” here. Indeed, the coherent states $|{\sf
p}/\lambda\rangle$ are not orthogonal for different values of parameter ${\sf
p}/\lambda$. Therefore, the vectors $|\psi_{[n]}^{\lambda}({\sf p})\rangle$
are not orthogonal for same values ${\sf p}$ and different values $\lambda$.

We will proceed as follows. Let the vector $|\psi_{[n]}^{\star}({\sf
p})\rangle\in\boldsymbol{H}^{\,\prime}$ will be the vector (47) where the
value $\lambda=\lambda_{\,0}=1/\pi$:

$|\psi_{[n]}^{\star}({\sf p})\rangle=C_{[n]}|{\sf
p}\rangle|n_{1},\dots,n_{k};{\sf p}/\lambda_{\,0}\rangle\,,\qquad
k=0,1,2,\dots\,.$ (48)

The state (48) which corresponds $k=0$, we call as ”ground state” of our
theory. Recall that the value $\lambda=\lambda_{\,0}$ corresponds to the
circulation $\Gamma_{0}$ of the unperturbed vortex ring:
$\Gamma_{0}={R_{0}^{\,2}}/{\pi t_{0}}$.

We declare that the physical subspace $\boldsymbol{H}_{phys}$ is spanned by
the following vectors

$|\psi_{phys}\rangle=\sum_{n_{1},\dots,n_{k}\atop n_{j}>1}\int
dp_{x}dp_{y}\,\varphi_{n_{1},\dots,n_{k}}(p_{x},p_{y})|\psi_{[n]}^{\star}({\sf
p})\rangle\,,$ (49)

where the wave functions $\varphi_{n_{1},\dots,n_{k}}(p_{x},p_{y})$ are
normalized.

Does the constructed space $\boldsymbol{H}_{phys}$ describes the states with
circulation $\Gamma_{0}$ only? In our opinion, the properties of coherent
states allow us to assume that this is not the case. Indeed, every system of
coherent states $|\alpha\rangle$ is an overdetermined system. Because
$\langle\alpha_{1}|\alpha_{2}\rangle\not=0$ even for different complex numbers
$\alpha_{1}$ and $\alpha_{2}$, we can conclude that any specific coherent
state $|\alpha_{0}\rangle$ contains ”some part” all other coherent states
$|\alpha\rangle$, $\alpha\not=\alpha_{0}$.

Returning to our notation, we have the following expression for the amplitude
for any $\lambda\in{\sf R}$:

$\langle\psi_{[n]}^{\lambda}({\sf
p})|\psi_{phys}\rangle=\varphi_{[n]}(p_{x},p_{y})\exp\left[-\frac{|{\sf
p}|^{2}R_{\,0}}{8\,\hbar\,p_{\,0}}\left(\frac{\lambda_{\,0}}{\lambda}-1\right)^{\\!2}\right]\,.$
(50)

Although $|\psi_{[n]}^{\lambda}({\sf p})\rangle\not\in\boldsymbol{H}_{phys}$
for the case $\lambda\not=1/\pi$, we suppose that the amplitude (50) defines
the probability density to find our dynamical system with circulation
$\Gamma={\lambda R_{0}^{2}}/{t_{0}}$, transverse impulse ${\sf
p}=p_{x}+ip_{y}$ and quantum numbers $[n]=\\{n_{1},\dots,n_{k}\\}$.

The numbers $[n]$ define the energy of our system. Indeed, in accordance with
our classical formulas for the Hamilton function, the quantum expression for
the Hamiltonian $\hat{H}$ takes the form

$\hat{H}=\frac{\hat{{\sf p}}^{+}\hat{{\sf
p}}}{2m_{0}}+\frac{\hbar}{t_{0}}\sum_{n>1}\hat{a}^{+}_{n}\hat{a}_{n}n\sqrt{n^{2}-1}\,.$

The following statement can be proved by direct verification:

###### Proposition 5

The vectors $|\psi_{[n]}^{\star}({\sf p})\rangle=C_{[n]}|{\sf
p}\rangle|n_{1},\dots,n_{k};{\sf p}/\lambda_{\,0}\rangle$ are the eigenvectors
of operator $\hat{H}$ with eigenvalues

${\mathcal{E}}=\frac{|{\sf
p}|^{2}}{2m_{0}}+\frac{\hbar}{t_{0}}\sum_{n>1}\sum_{j=1}^{k}\delta_{n,n_{j}}n\sqrt{n^{2}-1}\,.$
(51)

As regards the constraint in the form (32), the following equality takes
place:

$\langle\psi_{phys}|{\widehat{\Phi}_{0}}|\psi_{phys}\rangle=0\,.$

As a result, the constructed quantum description of the system allows for a
visual interpretation of the vortex as a structured particle. The connection
between external degrees of freedom (space $\boldsymbol{H}_{2}$) and internal
degrees of freedom (space $\boldsymbol{H}_{F}$) is nontrivial due to the
constraint (43). As a consequence, the quantum states of such a system are
entangled states. The formula (49) demonstrates that the vectors

$|\psi_{[0]}^{\star}({\sf p})\rangle=|{\sf p}\rangle|{\sf
p}/\lambda_{\,0}\rangle\,$

forms the ”set of the ground states” of our system. These vectors corresponds
to the vortex ring with the exact value of the transverse momentum ${\sf p}$,
the most probable value of the circulation $\Gamma_{0}={R_{0}^{\,2}}/{\pi
t_{0}}$ and the minimal energy ${|{\sf p}|^{2}}/{2m_{0}}$.

##  5 Concluding remarks

This paper has constructed a model describing the classical and quantum
dynamics of small perturbations of the vortex ring, which evolves according to
the LIE equation. The theory has three-dimensional constants: $R_{0}$ (radius
of unperturbed vortex ring), $p_{0}$ (momentum of unperturbed vortex ring),
and $m_{0}$ (the central charge for the central extension of Galilei group).
We quantized our model as an abstract dynamical system, without any connection
with the quantum properties of the surrounded liquid. Of course, taking into
account such property is an important area that is presented in the literature
(see [20], for example). In the case under consideration, when
$\boldsymbol{H}_{2}=L^{2}({\sf R}_{2})$, the eigenvalues of the momentum
operator belong to a continuous set. The same can be said about the energy
${\mathcal{E}}$ (see (51)) and the circulation $\Gamma$ (see (50)). However,
there is no contradiction with the experiment: for example, all the
experiments, where the quantization of the circulation was observed [7],
correspond to one of the real cases when certain boundary conditions are
present. To take into account any boundary conditions in our approach, we
should consider the case when $\boldsymbol{H}_{2}=L^{2}({\sf D})$, where the
domain ${\sf D}$ is a certain compact subset of a plane ${\sf R}_{2}$. We
suppose that corresponding theory leads to the discrete values of energy
${\mathcal{E}}$ and circulation $\Gamma$. The author hopes to devote the next
article to this issue.

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Overlap between usual and modified Bethe vectors

S. Belliard† and N. A. Slavnov‡<EMAIL_ADDRESS>nslavnov@mi-
ras.ru

† Institut Denis-Poisson, Université de Tours, Université d’Orléans, Parc de
Grammont, 37200 Tours, FRANCE

‡ Steklov Mathematical Institute of Russian Academy of Sciences,
8 Gubkina str., Moscow, 119991, RUSSIA

###### Abstract

We consider the overlap of Bethe vectors of the XXX spin chain with a diagonal
twist and the modified Bethe vectors with a general twist. We find a
determinant representation for this overlap under one additional condition on
the twist parameters. Such objects arise in the calculations of nonequilibrium
physics.

##  1 Introduction

Integrable quantum spin chains are a powerful tool for studying nonequilibrium
physics. They have parameters that can be considered time-dependent. A simple
example is an external magnetic field given by a Heaviside function that turns
the field on or off. One can also consider the change in the parameters that
define the boundary conditions. For example, a chain of $N$ spins can be
closed periodically or in the form of the Möbius strip. In the general case,
the classification of integrable boundary parameters follows from the Quantum
Inverse Scattering Method and related quantum algebras. In particular, the
twisting of isotropic closed spin chains is given by an arbitrary invertible
matrix.

To study the behavior of quantum integrable models when the boundary
conditions change, we should solve the eigenproblem for any integrable
boundary parameters and calculate the overlap between the states corresponding
to different values of these parameters. For this, we use the Quantum Inverse
Scattering Method [1, 2, 3]. In this method, the eigenvectors of quantum
Hamiltonians are constructed using the algebraic Bethe ansatz, and the roots
of the Bethe equations parameterize the spectrum. A modified version of the
Bethe ansatz is needed in the case of so-called ”off-diagonal boundaries” [4].
The spectrum of such models is given by modified Bethe equations, which
contain the inhomogeneous term [5, 6]. In this case, there is an additional
restriction on the number of Bethe equations.

In a recent series of papers, we studied the scalar products of modified Bethe
vectors [7, 8, 9]. We found that the well-known formulas of the standard
algebraic Bethe ansatz smoothly transform into a modified form that has the
same structure. This applies both to scalar products of vectors of a general
form [12] and to scalar products containing the eigenvectors of a modified
transfer matrix [10, 11, 12, 13]. In this paper, we consider the overlap
(scalar product) of two vectors corresponding to different transfer matrices.
One of the vectors corresponds to a transfer matrix with a diagonal twist. It
can be constructed within the framework of the standard algebraic Bethe
ansatz. The second vector is a modified Bethe vector, which corresponds to a
transfer matrix with a twist of the general form. To construct this vector,
one needs to use the modified Bethe ansatz. We show that when one additional
condition is imposed on the twist parameters, such an overlap has a compact
determinant representation. This opens a possibility to calculate quenches in
the XXX chains of an arbitrary spin.

The paper is organized as follows. In section 2, we recall basic notions of
the Quantum Inverse Scattering Method. Here we also introduce a special
notation used below. In section 3, we introduce a modified transfer matrix and
construct modified Bethe vectors. Section 4 is devoted to a modified Izergin
determinant [7]. Here we derive a new representation which contains a set of
arbitrary complex numbers. In section 5, we consider the overlap between Bethe
vectors with diagonal twist and modified Bethe vectors of general type. We
show that under one additional constraint between the twist parameters, this
overlap has a determinant representation. Appendix A contains some auxiliary
formulas for the modified Izergin determinant and proof of some propositions
used in the calculation.

##  2 Quantum inverse scattering method

To formulate the problem within the Quantum Inverse Scattering Method
framework, we consider a monodromy matrix

$T(u)=\begin{pmatrix}t_{11}(u)&t_{12}(u)\\\ t_{21}(u)&t_{22}(u)\end{pmatrix}.$
(2.1)

The matrix elements $t_{kl}(u)$ act in a Hilbert space $\mathcal{H}$ and
depend on a complex parameter $u$. The commutation relations between
$t_{kl}(u)$ are defined by an RTT-relation

$R_{12}(u-v)T_{1}(u)T_{2}(v)=T_{2}(v)T_{1}(u)R_{12}(u-v).$ (2.2)

Here $T_{1}(u)=T(u)\otimes\mathbf{1}$ and $T_{2}(u)=\mathbf{1}\otimes T(u)$,
where $\mathbf{1}$ is the identity matrix in $\mathbb{C}^{2}$. The $R$-matrix
$R_{12}(u)$ acts in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and has the form

$R(u)=\frac{u}{c}(\mathbf{1}\otimes\mathbf{1})+P.$ (2.3)

Here $P$ the permutation operator such that $Px\otimes y=y\otimes x$ for any
$x,y\in\mathbb{C}^{2}$, and $c$ is a constant. The $R$-matrix (2.3) has the
following property:

$[R_{12}(u-v),K_{1}K_{2}]=0,$ (2.4)

for any $2\times 2$ matrix $K$. This property leads to the fact that a matrix
$KT(u)$ satisfies the RTT-relation (2.2).

The trace of the monodromy matrix $\mathop{\rm tr}T(u)=t_{11}(u)+t_{22}(u)$ is
called a transfer matrix. Due to the RTT-relation (2.2),

$[\mathop{\rm tr}T(u),\mathop{\rm tr}T(v)]=0,$ (2.5)

for any complex $u$ and $v$. This property of the transfer matrix allows us to
consider it as a generating function of the integrals of motion of a quantum
model. Due to the cyclicity of the trace, this model has periodic boundary
conditions. The trace of the twisted monodromy matrix $\mathop{\rm tr}(KT(u))$
can also be used to generate integrals of motion. However, they satisfy
boundary conditions of a more general form.

Let us define a highest weight representation by
$V(\lambda_{1}(u),\lambda_{2}(u))$, where $\lambda_{i}(u)$ are some complex
valued functions, and the highest weight vector $|0\rangle\in\mathcal{H}$ is
defined by

$\displaystyle t_{ii}(u)|0\rangle=\lambda_{i}(u)|0\rangle,\qquad
t_{21}(u)|0\rangle=0.$ (2.6)

The action of the operator $t_{12}(u)$ on $|0\rangle$ is free. A state
obtained by the successive action of the $t_{12}$ operators on the highest
weight vector is called a Bethe vector

$\displaystyle|\Phi(\bar{u})\rangle=t_{12}(u_{1})\dots
t_{12}(u_{n})|0\rangle.$ (2.7)

Here $n=0,1,\dots$, and $\bar{u}=\\{u_{1},\dots,u_{n}\\}$. If the parameters
$\bar{u}$ are generic complex numbers, then $|\Phi(\bar{u})\rangle$ is called
the off-shell Bethe vector. Under certain constraint on the parameters
$\bar{u}$ (see section 3), the vector (2.7) becomes an eigenvector of the
transfer matrix. Then it is called the on-shell Bethe vector.

To study scalar products we also introduce a dual vector $\langle
0|\in\mathcal{H}^{*}$ defined by

$\displaystyle\langle 0|t_{ii}(u)=\lambda_{i}(u)\langle 0|,\qquad\langle
0|t_{12}(u)=0,$ (2.8)

and a condition $\langle 0|0\rangle=1$. The functions $\lambda_{i}(u)$ are the
same as in (2.6). Dual Bethe vectors are constructed by successive action of
the $t_{21}$ operators on $\langle 0|$:

$\langle\Phi(\bar{u})|=\langle 0|t_{21}(u_{1})\dots t_{21}(u_{n}).$ (2.9)

###  2.1 Notation

To simplify formulas, we introduce three rational functions

$\displaystyle g(u,v)=\frac{c}{u-v},\quad
f(u,v)=1+g(u,v)=\frac{u-v+c}{u-v},\quad
h(u,v)=\frac{f(u,v)}{g(u,v)}=\frac{u-v+c}{c},$ (2.10)

where $c$ is the constant entering the $R$-matrix. It is easy to see that the
functions (2.10) have the following properties:

$\displaystyle\chi(u,v)\Bigr{|}_{c\to-c}=\chi(v,u),\qquad\chi(-u,-v)=\chi(v,u),\qquad\chi(u-c,v)=\chi(u,v+c),$
(2.11)

where $\chi$ is any of the three functions. Besides, we have

$\displaystyle g(u,v-c)=\frac{1}{h(u,v)},\quad h(u,v+c)=\frac{1}{g(u,v)},\quad
f(u,v+c)=\frac{1}{f(v,u)}.$ (2.12)

Below we deal with sets of complex parameters. We denote them by a bar (see,
e.g., (2.7)), for example, $\bar{u}=\\{u_{1},\dots,u_{n}\\}$. Notation
$\bar{u}\pm c$ means that $\pm c$ is added to all the arguments of the set
$\bar{u}$. The notation $\bar{u}_{k}$ refers to the subset
$\bar{u}_{k}=\bar{u}\setminus u_{k}$.

To make the formulas more compact, we use a shorthand notation for the
products of the rational functions (2.10), the operators $t_{kl}(u)$ (2.1),
and their vacuum eigenvalues $\lambda_{i}(u)$ (2.6). Namely, if the function
(operator) depends on a set of variables, then this means the product over the
corresponding set. For example, if $\bar{u}=\\{u_{1},\dots,u_{n}\\}$, then

$t_{kl}(\bar{u})=\prod_{j=1}^{n}t_{kl}(u_{j}),\quad\lambda_{i}(\bar{u})=\prod_{j=1}^{n}\lambda_{i}(u_{j}),\quad
f(z,\bar{u})=\prod_{j=1}^{n}f(z,u_{j}),\quad
f(\bar{u}_{k},u_{k})=\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}f(u_{j},u_{k}),$ (2.13)

and so on. Note that the RTT-relation (2.2) implies $[t_{kl}(u),t_{kl}(v)]=0$.
Thus, the first product in (2.13) is well defined. Later we will extend this
convention to the products of matrix elements of the twisted monodromy matrix.

Notation $f(\bar{u},\bar{v})$ means the double product over the sets $\bar{u}$
and $\bar{v}$. By definition, any product over the empty set is equal to $1$.
A double product is equal to $1$ if at least one of the sets is empty.

Finally, for any set of complex parameters $\bar{u}=\\{u_{1},\dots,u_{n}\\}$
such that $n\geq 2$, we also introduce special products of the $g$-functions

$\Delta(\bar{u})=\prod_{1\leq k<j\leq
n}g(u_{j},u_{k}),\qquad\Delta^{\prime}(\bar{u})=\prod_{1\leq k<j\leq
n}g(u_{k},u_{j}).$ (2.14)

It is easy to see that
$\Delta(\bar{u})=(-1)^{n(n-1)/2}\Delta^{\prime}(\bar{u})$. For $n=0,1$, we set
$\Delta(\bar{u})=\Delta^{\prime}(\bar{u})=1$ by definition.

##  3 Twisted monodromy matrix

We consider two twisted monodromy matrices $K^{(\ell)}T(u)$, $\ell=1,2$.

The first one corresponds to the twist matrix $K^{(1)}=\mathop{\rm
diag}(1,\alpha)$, where $\alpha$ is a complex parameter. Then the twisted
transfer matrix has the form

$t^{(1)}(u)=t_{11}(u)+\alpha t_{22}(u).$ (3.1)

We call it a diagonal transfer matrix. It generates a model with quasi-
periodic boundary conditions. The eigenvectors (resp. dual eigenvectors) of
the diagonal transfer matrix have the form (2.7) (resp. (2.9))

$\displaystyle|\Phi^{(1)}(\bar{u})\rangle=t_{12}(\bar{u})|0\rangle,\qquad\langle\Phi^{(1)}(\bar{u})|=\langle
0|t_{21}(\bar{u}).$ (3.2)

We have added an extra superscript to the vectors to emphasize that they refer
to the transfer matrix $t^{(1)}(u)$.

If the parameters $\bar{u}=\\{u_{1},\dots,u_{n}\\}$ are arbitrary complex
numbers, then the vectors (3.2) are off-shell. If these parameters satisfy
twisted Bethe equations

$\lambda_{1}(u_{j})f(\bar{u}_{j},u_{j})=\alpha\lambda_{2}(u_{j})f(u_{j},\bar{u}_{j}),\qquad
j=1,\dots,n,$ (3.3)

then

$\displaystyle
t^{(1)}(v)|\Phi^{(1)}(\bar{u})\rangle=\Lambda^{(1)}(v|\bar{u})|\Phi^{(1)}(\bar{u})\rangle,\qquad\langle\Phi^{(1)}(\bar{u})|t^{(1)}(v)=\Lambda^{(1)}(v|\bar{u})\langle\Phi^{(1)}(\bar{u})|,$
(3.4)

where

$\Lambda^{(1)}(v|\bar{u})=\lambda_{1}(v)f(\bar{u},v)+\alpha\lambda_{2}(v)f(v,\bar{u}).$
(3.5)

The vectors $|\Phi^{(1)}(\bar{u})\rangle$ and $\langle\Phi^{(1)}(\bar{u})|$
are called on-shell provided the conditions (3.3) are fulfilled. In the case
of the XXX spin-$1/2$ chain, the number of parameters
$\\{u_{1},\dots,u_{n}\\}$ does not exceed the length of the chain.

The second deformation corresponds to the twist matrix of the general form

$\displaystyle K^{(2)}=\begin{pmatrix}\tilde{\kappa}&\;\kappa^{+}\\\
\;\kappa^{-}&\kappa\end{pmatrix},$ (3.6)

where the entries of $K^{(2)}$ are arbitrary complex numbers. This twist
produces models with non-diagonal boundary conditions. The standard procedure
of the algebraic Bethe ansatz should be modified in this case [14, 8]. We
present the twist matrix in the form $K^{(2)}=BDA$ where

$A=\sqrt{\mu}\begin{pmatrix}1&\frac{\rho_{2}}{\kappa^{-}}\\\
\frac{\rho_{1}}{\kappa^{+}}&1\end{pmatrix},\qquad
B=\sqrt{\mu}\begin{pmatrix}1&\frac{\rho_{1}}{\kappa^{-}}\\\
\frac{\rho_{2}}{\kappa^{+}}&1\end{pmatrix},\qquad
D=\begin{pmatrix}\tilde{\kappa}-\rho_{1}&0\\\ 0&\kappa-\rho_{2}\end{pmatrix}.$
(3.7)

Here

$\mu=\frac{1}{1-\frac{\rho_{1}\rho_{2}}{\kappa^{+}\kappa^{-}}},$ (3.8)

and the parameters $\rho_{\ell}$ satisfy the following condition:

$\displaystyle\rho_{1}\rho_{2}-(\kappa\rho_{1}+\tilde{\kappa}\rho_{2})+\kappa^{+}\kappa^{-}=0.$
(3.9)

Then a twisted transfer matrix is defined as follows:

$t^{(2)}(u)=\mathop{\rm tr}(K^{(2)}T(u))=\mathop{\rm tr}(D\overline{T}(u)).$
(3.10)

Here $\overline{T}(u)$ is a modified monodromy matrix

$\displaystyle\overline{T}(u)=AT(u)B=\begin{pmatrix}\nu_{11}(u)&\nu_{12}(u)\\\
\nu_{21}(u)&\nu_{22}(u)\end{pmatrix}.$ (3.11)

The entries of $\overline{T}(u)$ have the following expressions in terms of
the initial monodromy matrix elements:

$\displaystyle\nu_{11}(z)=\mu\Big{(}t_{11}(z)+\frac{\rho_{2}}{\kappa^{+}}t_{12}(z)+\frac{\rho_{2}}{\kappa^{-}}t_{21}(z)+\frac{\rho_{2}^{2}}{\kappa^{-}\kappa^{+}}t_{22}(z)\Big{)},$
(3.12)
$\displaystyle\nu_{22}(z)=\mu\Big{(}t_{22}(z)+\frac{\rho_{1}}{\kappa^{+}}t_{12}(z)+\frac{\rho_{1}}{\kappa^{-}}t_{21}(z)+\frac{\rho_{1}^{2}}{\kappa^{-}\kappa^{+}}t_{11}(z)\Big{)},$
(3.13)
$\displaystyle\nu_{12}(z)=\mu\Big{(}t_{12}(z)+\frac{\rho_{1}}{\kappa^{-}}t_{11}(z)+\frac{\rho_{2}}{\kappa^{-}}t_{22}(z)+\frac{\rho_{1}\rho_{2}}{(\kappa^{-})^{2}}t_{21}(z)\Big{)},$
(3.14)
$\displaystyle\nu_{21}(z)=\mu\Big{(}t_{21}(z)+\frac{\rho_{1}}{\kappa^{+}}t_{11}(z)+\frac{\rho_{2}}{\kappa^{+}}t_{22}(z)+\frac{\rho_{1}\rho_{2}}{(\kappa^{+})^{2}}t_{12}(z)\Big{)}.$
(3.15)

The modified Bethe vectors (resp. the dual modified Bethe vectors) related to
the twisted transfer matrix (3.10) can be constructed from the new entries
$\nu_{12}(z)$ (resp. $\nu_{21}(z)$). They are given by

$\displaystyle|\Phi^{(2)}(\bar{u})\rangle=\nu_{12}(\bar{u})|0\rangle,\qquad\langle\Phi^{(2)}(\bar{u})|=\langle
0|\nu_{21}(\bar{u}).$ (3.16)

Here we have used the convention on the shorthand notation for the products of
the operators $\nu_{ij}(u)$.

If $\bar{u}=\\{u_{1},\dots,u_{N}\\}$ is a set of arbitrary complex numbers,
then vectors (3.16) are off-shell. However, if the set $\bar{u}$ satisfies
inhomogeneous Bethe equations

$\displaystyle(\tilde{\kappa}-\rho_{1})\lambda_{1}(u_{j})f(\bar{u}_{j},u_{j})=(\kappa-\rho_{2})\lambda_{2}(u_{j})f(u_{j},\bar{u}_{j})+(\rho_{1}+\rho_{2})\lambda_{1}(u_{j})\lambda_{2}(u_{j})g(u_{j},\bar{u}_{j}),$
(3.17)

for $j=1,\dots,N$, then they are eigenvectors of the twisted transfer matrix
(3.10)

$\displaystyle
t^{(2)}(v)|\Phi^{(2)}(\bar{u})\rangle=\Lambda^{(2)}(v|\bar{u})|\Phi^{(2)}(\bar{u})\rangle,\qquad\langle\Phi^{(2)}(\bar{u})|t^{(2)}(v)=\Lambda^{(2)}(v|\bar{u})\langle\Phi^{(2)}(\bar{u})|,$
(3.18)

with an eigenvalue

$\displaystyle\Lambda^{(2)}(v|\bar{u})=(\tilde{\kappa}-\rho_{1})\lambda_{1}(v)f(\bar{u},v)+(\kappa-\rho_{2})\lambda_{2}(v)f(v,\bar{u})+(\rho_{1}+\rho_{2})\lambda_{1}(v)\lambda_{2}(v)g(v,\bar{u}).$
(3.19)

In this case, we call the vectors $|\Phi^{(2)}(\bar{u})\rangle$ and
$\langle\Phi^{(2)}(\bar{u})|$ modified on-shell vectors.

The number of parameters $\\{u_{1},\dots,u_{N}\\}$ depends on the chain length
and spin. In particular, in the XXX spin-$1/2$ chain, $N$ coincides with the
number of sites. In the case of higher spins, $N$ exceeds the chain length
(see [8] for more details).

The main goal of this paper is to calculate an overlap

$\displaystyle
S^{n,N}(\bar{v},\bar{u})=\langle\Phi^{(1)}(\bar{v})|\Phi^{(2)}(\bar{u})\rangle.$
(3.20)

Here $\langle\Phi^{(1)}(\bar{v})|$ and $|\Phi^{(2)}(\bar{u})\rangle$ are the
eigenvectors of the corresponding transfer matrices, $n=\\#\bar{v}$, and
$N=\\#\bar{u}$. Then according to Fermi’s Golden rule for the transition
between the two twists, we obtain

$\displaystyle\Gamma_{1\to
2}=\frac{2\pi}{\hbar}\Big{|}\langle\Phi^{(1)}(\bar{v})|(H^{(1)}-H^{(2)})|\Phi^{(2)}(\bar{u})\rangle\Big{|}^{2}{\rho^{(2)}}(\bar{u}).$
(3.21)

Here $H^{(1)}$ and $H^{(2)}$ respectively are the Hamiltonians corresponding
to the transfer matrices $t^{(1)}$ and $t^{(2)}$, and $\rho^{(2)}(\bar{u})$ is
a density of the resulting states.

In the case of the XXX spin-$1/2$ chains we can express Hamiltonians from the
twisted transfer matrix using

$H^{(1)}-H^{(2)}=2c\frac{d}{dz}\big{(}\log t^{(1)}(z)-\log
t^{(2)}(z)\big{)}\Bigr{|}_{z=0}.$ (3.22)

Thus, we find

$\displaystyle\Gamma_{1\to
2}=\frac{2\pi}{\hbar}\left|2c\frac{d}{dz}\log\left(\frac{\Lambda^{(1)}(z|\bar{v})}{\Lambda^{(2)}(z|\bar{u})}\right)\right|^{2}_{z=0}\big{|}S^{n,N}(\bar{v},\bar{u})\big{|}^{2}{\rho}^{(2)}(\bar{u}).$
(3.23)

##  4 Modified Izergin determinant

In the formulas for the scalar products, a modified Izergin determinant (MID)
arises.

######  Definition 4.1.

Let $\bar{u}=\\{u_{1},\dots,u_{n}\\}$, $\bar{v}=\\{v_{1},\dots,v_{m}\\}$, and
$z$ be arbitrary complex numbers. Then the MID
$K_{n,m}^{(z)}(\bar{u}|\bar{v})$ is defined by

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=\det_{m}\left(-z\delta_{jk}+\frac{f(\bar{u},v_{j})f(v_{j},\bar{v}_{j})}{h(v_{j},v_{k})}\right).$
(4.1)

Alternatively the MID can be presented as

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=(1-z)^{m-n}\det_{n}\left(\delta_{jk}f(u_{j},\bar{v})-z\frac{f(u_{j},\bar{u}_{j})}{h(u_{j},u_{k})}\right).$
(4.2)

The proof of the equivalence of representations (4.1) and (4.2) can be found
in proposition 4.1 of [15].

For $m=n$ and $z=1$, the MID turns into the usual Izergin determinant [16].
The latter is equal to the partition function of the six-vertex model with
domain wall boundary condition [12, 16].

We will also use a conjugated MID

$\overline{K}_{n,m}^{(z)}(\bar{u}|\bar{v})=K_{n,m}^{(z)}(\bar{u}|\bar{v})\Bigr{|}_{c\to-c}=\det_{m}\left(-z\delta_{jk}+\frac{f(v_{j},\bar{u})f(\bar{v}_{j},v_{j})}{h(v_{k},v_{j})}\right).$
(4.3)

Equivalently, it can be defined as follows:

$\overline{K}_{n,m}^{(z)}(\bar{u}|\bar{v})=(1-z)^{m-n}\det_{n}\left(\delta_{jk}f(\bar{v},u_{j})-z\frac{f(\bar{u}_{j},u_{j})}{h(u_{k},u_{j})}\right).$
(4.4)

There exist also representations for MID, which contain additional parameters.

######  Proposition 4.1.

Let $\bar{u}=\\{u_{1},\dots,u_{n}\\}$,
$\bar{\eta}=\\{\eta_{1},\dots,\eta_{n}\\}$, $\bar{v}=\\{v_{1},\dots,v_{m}\\}$,
and $z$ be arbitrary complex numbers. Then

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=(1-z)^{m-n}\Delta^{\prime}(\bar{u})\Delta(\bar{\eta})\det_{n}\left(\frac{f(u_{j},\bar{v})}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right).$
(4.5)

Proof. Let $\bar{u}=\\{u_{1},\dots,u_{n}\\}$ be pair-wise distinct complex
numbers. Consider an $n\times n$ matrix $W$ with the elements

$W_{jk}=\frac{g(u_{j},\bar{u}_{j})}{g(u_{j},\bar{\eta}_{k})}.$ (4.6)

Here $\bar{\eta}=\\{\eta_{1},\dots,\eta_{n}\\}$ are arbitrary pair-wise
distinct complex numbers. The entries $W_{jk}$ are proportional to the Cauchy
matrix $g(u_{j},\eta_{k})$. Thus,

$\det_{n}W=\frac{\Delta^{\prime}(\bar{u})\Delta(\bar{u})}{g(\bar{u},\bar{\eta})}\det_{n}g(u_{j},\eta_{k})=\frac{\Delta(\bar{u})}{\Delta(\bar{\eta})}.$
(4.7)

Thus, the determinant of $W$ exists and is non-vanishing. We take
representation (4.2) and transform it as follows:

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=\frac{(1-z)^{m-n}}{\det_{n}W}\det_{n}\left(f(u_{j},\bar{v})W_{jk}-z\sum_{l=1}^{n}\frac{f(u_{j},\bar{u}_{j})}{h(u_{j},u_{l})}W_{lk}\right).$
(4.8)

The sum over $l$ is easily computable. Indeed, let

$G_{jk}=\sum_{\ell=1}^{n}\frac{W_{lk}}{h(u_{j},u_{l})}=\sum_{\ell=1}^{n}\frac{g(u_{\ell},\eta_{k})}{h(u_{j},u_{\ell})}\frac{g(u_{\ell},\bar{u}_{\ell})}{g(u_{\ell},\bar{\eta})}.$
(4.9)

Then we have

$\frac{1}{2\pi
ic}\oint_{|z|=R\to\infty}\frac{g(z,\eta_{k})}{h(u_{j},z)}\frac{g(z,\bar{u})}{g(z,\bar{\eta})}\,dz=0=G_{jk}-\frac{h(u_{j},\bar{\eta}_{k})}{h(u_{j},\bar{u})}.$
(4.10)

Substituting this into (4.8) we obtain

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=\frac{(1-z)^{m-n}\Delta(\bar{\eta})}{\Delta(\bar{u})}\det_{n}\left(f(u_{j},\bar{v})\frac{g(u_{j},\bar{u}_{j})}{g(u_{j},\bar{\eta}_{k})}-zg(u_{j},\bar{u}_{j})h(u_{j},\bar{\eta}_{k})\right).$
(4.11)

Extracting the products $g(u_{j},\bar{u}_{j})$ we arrive at (4.5). It remains
to note that the limits $u_{j}=u_{k}$ and $\eta_{j}=\eta_{k}$
($j,k=1,\dots,n$) are well defined. Indeed, in this case, the prefactor
$\Delta^{\prime}(\bar{u})\Delta(\bar{\eta})$ has a pole, while the determinant
vanishes. Therefore, representation (4.5) is valid for any complex $\bar{u}$
and $\bar{\eta}$.

Similarly, one can prove the following representation for the MID:

$K_{n,m}^{(z)}(\bar{u}|\bar{v})=\Delta^{\prime}(\bar{u})\Delta(\bar{\eta})\det_{m}\left(f(\bar{u},v_{j})h(u_{j},\bar{\eta}_{k})-\frac{z}{g(u_{j},\bar{\eta}_{k})}\right).$
(4.12)

Here $\bar{\eta}=\\{\eta_{1},\dots,\eta_{m}\\}$ is a set of arbitrary complex
numbers.

######  Corollary 4.1.

Let $\bar{u}=\\{u_{1},\dots,u_{n}\\}$,
$\bar{\eta}=\\{\eta_{1},\dots,\eta_{n}\\}$, and $z$ be arbitrary complex
numbers. Then

$\Delta^{\prime}(\bar{u})\Delta(\bar{\eta})\det_{n}\left(\frac{1}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right)=(1-z)^{n}.$
(4.13)

Proof. Setting $\bar{v}=\emptyset$ in (4.5) we arrive at

$\Delta^{\prime}(\bar{u})\Delta(\bar{\eta})\det_{n}\left(\frac{1}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right)=(1-z)^{n}K_{n,0}^{(z)}(\bar{u}|\emptyset).$
(4.14)

Then (4.13) follows from (4.1).

Some other properties of the MID are given in appendix A (see also [7] for a
more detailed description).

##  5 Overlap

Let us now proceed to the calculation of the overlap (3.20)

$\displaystyle S^{n,N}(\bar{v},\bar{u})=\langle
0|t_{21}(\bar{v})\nu_{12}(\bar{u})|0\rangle.$ (5.1)

Recall that here $n=\\#\bar{v}$ and $N=\\#\bar{u}$. Ultimately, we are
interested in the case when the parameters $\bar{v}$ and $\bar{u}$
respectively satisfy equations (3.3) and (3.17). However, in this paper, we
consider a more general case where the parameters $\bar{u}$ are arbitrary
complex numbers. In a particular, they can satisfy the system (3.17).
Moreover, at the first stage of calculations, we also do not impose any
restrictions on the parameters $\bar{v}$. We only require that $n\leq N$,
because otherwise, the scalar product (5.1) vanishes.

###  5.1 Scalar product of off-shell vectors

It follows from (3.14) that the vector $\mu^{-N}|\Phi^{(2)}(\bar{u})\rangle$
does not depend on $\kappa^{+}$. Therefore, the scalar product
$\mu^{-N}S^{n,N}(\bar{v},\bar{u})$ also does not depend on $\kappa^{+}$. On
the other hand, it follows from (3.15) that

$t_{21}(v)=\lim_{\kappa^{+}\to\infty}\nu_{21}(v).$ (5.2)

Thus,

$\displaystyle\langle
0|t_{21}(\bar{v})\nu_{12}(\bar{u})|0\rangle=\mu^{N}\lim_{\kappa^{+}\to\infty}\langle
0|\nu_{21}(\bar{v})\nu_{12}(\bar{u})|0\rangle,$ (5.3)

where we used $\mu\to 1$ as $\kappa^{+}\to\infty$. In its turn, a formula for
the scalar product $\langle 0|\nu_{21}(\bar{v})\nu_{12}(\bar{u})|0\rangle$ was
derived in [7]:

$\langle
0|\nu_{21}(\bar{v})\nu_{12}(\bar{u})|0\rangle=\mu^{N+n}\left(\frac{\rho_{1}}{\kappa^{-}}\right)^{N-n}\sum_{\begin{subarray}{c}\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}\\\
\\{\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{v}\end{subarray}}\left(\frac{\rho_{1}}{\rho_{2}}\right)^{n_{\scriptscriptstyle\rm
II}-N_{\scriptscriptstyle\rm II}}\lambda_{2}(\bar{v}_{\scriptscriptstyle\rm
I})\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{v}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{u}_{\scriptscriptstyle\rm I})\\\ \times
f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})K^{(1/\mu)}_{N_{\scriptscriptstyle\rm II},n_{\scriptscriptstyle\rm
II}}(\bar{u}_{\scriptscriptstyle\rm II}|\bar{v}_{\scriptscriptstyle\rm
II})\overline{K}^{(1/\mu)}_{N_{\scriptscriptstyle\rm
I},n_{\scriptscriptstyle\rm I}}(\bar{u}_{\scriptscriptstyle\rm
I}|\bar{v}_{\scriptscriptstyle\rm I}).$ (5.4)

Here the sum is taken over all possible partitions
$\\{\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{v}$ and $\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}$ without restrictions on
the cardinalities of the subsets $N_{\scriptscriptstyle\rm
I,\scriptscriptstyle\rm II}=\\#\bar{u}_{\scriptscriptstyle\rm
I,\scriptscriptstyle\rm II}$ and $n_{\scriptscriptstyle\rm
I,\scriptscriptstyle\rm II}=\\#\bar{v}_{\scriptscriptstyle\rm
I,\scriptscriptstyle\rm II}$.

Taking the limit $\kappa^{+}\to\infty$ in (5.4) corresponds to the limit
$\mu\to 1$. Thus, we obtain

$S^{n,N}(\bar{v},\bar{u})=\mu^{N}\left(\frac{\rho_{1}}{\kappa^{-}}\right)^{N-n}\sum_{\begin{subarray}{c}\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}\\\
\\{\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{v}\end{subarray}}\left(\frac{\rho_{1}}{\rho_{2}}\right)^{n_{\scriptscriptstyle\rm
II}-N_{\scriptscriptstyle\rm II}}\lambda_{2}(\bar{v}_{\scriptscriptstyle\rm
I})\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{v}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{u}_{\scriptscriptstyle\rm I})\\\ \times
f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})K^{(1)}_{N_{\scriptscriptstyle\rm II},n_{\scriptscriptstyle\rm
II}}(\bar{u}_{\scriptscriptstyle\rm II}|\bar{v}_{\scriptscriptstyle\rm
II})\overline{K}^{(1)}_{N_{\scriptscriptstyle\rm I},n_{\scriptscriptstyle\rm
I}}(\bar{u}_{\scriptscriptstyle\rm I}|\bar{v}_{\scriptscriptstyle\rm I}).$
(5.5)

The sum is taken over partitions described above. Formally, we still have no
restrictions on the cardinalities of the subsets. However, the two MID in
(5.5) vanish if $N_{\scriptscriptstyle\rm II}<n_{\scriptscriptstyle\rm II}$ or
$N_{\scriptscriptstyle\rm I}<n_{\scriptscriptstyle\rm I}$ (see (4.1), (4.4)).

###  5.2 Scalar product with on-shell Bethe vector

Up to now both vectors were off-shell. Let $\langle\Phi^{(1)}(\bar{v})|$ be
on-shell Bethe vector, that means that the set $\bar{v}$ satisfies Bethe
equations (3.3). Now we can substitute the Bethe equations in the form

$\lambda_{1}(\bar{v}_{\scriptscriptstyle\rm
II})f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})=\alpha^{n_{\scriptscriptstyle\rm
II}}\lambda_{2}(\bar{v}_{\scriptscriptstyle\rm
I})f(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm I}).$
(5.6)

Then we obtain

$S^{n,N}(\bar{v},\bar{u})=\mu^{N}\left(\frac{\rho_{1}}{\kappa^{-}}\right)^{N-n}\lambda_{2}(\bar{v})\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}\left(\frac{\rho_{2}}{\rho_{1}}\right)^{N_{\scriptscriptstyle\rm
II}}\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})G(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm II}),$
(5.7)

where

$G(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=\sum_{\\{\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{v}}\left(\frac{\alpha\rho_{1}}{\rho_{2}}\right)^{n_{\scriptscriptstyle\rm
II}}f(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})K^{(1)}_{N_{\scriptscriptstyle\rm II},n_{\scriptscriptstyle\rm
II}}(\bar{u}_{\scriptscriptstyle\rm II}|\bar{v}_{\scriptscriptstyle\rm
II})\overline{K}^{(1)}_{N_{\scriptscriptstyle\rm I},n_{\scriptscriptstyle\rm
I}}(\bar{u}_{\scriptscriptstyle\rm I}|\bar{v}_{\scriptscriptstyle\rm I}).$
(5.8)

Let us impose an additional constraint on the twist parameters:
$\alpha=-\rho_{2}/\rho_{1}$. Then the sum over partitions in (5.8) can be
explicitly computed via (A.6):

$G(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})\Bigr{|}_{\alpha=-\rho_{2}/\rho_{1}}=(-1)^{n}f(\bar{v},\bar{u}_{\scriptscriptstyle\rm
I})K^{(1)}_{N,n}(\\{\bar{u}_{\scriptscriptstyle\rm
I}-c,\bar{u}_{\scriptscriptstyle\rm II}\\}|\bar{v}).$ (5.9)

Respectively, the scalar product takes the form

$S^{n,N}(\bar{v},\bar{u})=(-1)^{n}\mu^{N}\left(\frac{\rho_{1}}{\kappa^{-}}\right)^{N-n}\lambda_{2}(\bar{v})\\\
\times\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}\left(\frac{\rho_{2}}{\rho_{1}}\right)^{N_{\scriptscriptstyle\rm
II}}\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{v},\bar{u}_{\scriptscriptstyle\rm
I})K^{(1)}_{N,n}(\\{\bar{u}_{\scriptscriptstyle\rm
I}-c,\bar{u}_{\scriptscriptstyle\rm II}\\}|\bar{v}).$ (5.10)

The sum over partitions of the set $\bar{u}$ in (5.10) is computed via
proposition A.3. For this, it is enough to use representation for MID in the
form (4.5)

$K_{N,n}^{(1)}(\bar{w}|\bar{v})=\lim_{z\to
1}(1-z)^{n-N}\Delta(\bar{w})\Delta^{\prime}(\bar{\eta})\det_{N}\left(\frac{f(w_{j},\bar{v})}{g(w_{j},\bar{\eta}_{k})}-zh(w_{j},\bar{\eta}_{k})\right),$
(5.11)

where $\bar{w}=\\{\bar{u}_{\scriptscriptstyle\rm
I}-c,\bar{u}_{\scriptscriptstyle\rm II}\\}$ for any fixed partition
$\\{\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}$. Then we satisfy the condition of proposition A.3. Recall
also that $\bar{\eta}=\\{\eta_{1},\dots,\eta_{N}\\}$ are arbitrary complex
numbers. Thus, we find

$S^{n,N}(\bar{v},\bar{u})=(-1)^{n}\mu^{N}\left(\frac{\kappa^{-}}{\rho_{1}}\right)^{n}\lambda_{2}(\bar{v})\Delta^{\prime}(\bar{\eta})\Delta(\bar{u})\lim_{z\to
1}(1-z)^{n-N}\det_{N}\mathcal{N}_{jk}(z),$ (5.12)

where

$\mathcal{N}_{jk}(z)=(-1)^{N-1}\hat{\lambda}_{1}(u_{j})\left(h(\bar{\eta}_{k},u_{j})-z\frac{f(\bar{v},u_{j})}{g(\bar{\eta}_{k},u_{j})}\right)+\hat{\lambda}_{2}(u_{j})\left(\frac{f(u_{j},\bar{v})}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right),$
(5.13)

and

$\hat{\lambda}_{\ell}(u)=\frac{\rho_{\ell}}{\kappa^{-}}\lambda_{\ell}(u),\qquad\ell=1,2.$
(5.14)

###  5.3 How to take the limit $z\to 1$

To take the limit $z\to 1$ in equation (5.12), we set $\eta_{k}=v_{k}$ for
$k=1,\dots,n$ in the matrix (5.13). Then

$S^{n,N}(\bar{v},\bar{u})=(-1)^{n}\mu^{N}\left(\frac{\kappa^{-}}{\rho_{1}}\right)^{n}\lambda_{2}(\bar{v})\Delta^{\prime}(\bar{v})\Delta^{\prime}(\bar{\eta})g(\bar{v},\bar{\eta})\Delta(\bar{u})\lim_{z\to
1}(1-z)^{n-N}\det_{N}\mathcal{N}_{jk}(z),$ (5.15)

where now and further $\bar{\eta}=\\{\eta_{n+1},\dots,\eta_{N}\\}$. The matrix
$\mathcal{N}_{jk}(z)$ now consists of two parts:

$\displaystyle\mathcal{N}_{jk}(z)=\mathcal{N}_{jk}^{(1)}(z),\qquad
k=1,\dots,n,$ (5.16)
$\displaystyle\mathcal{N}_{jk}(z)=\mathcal{N}_{jk}^{(2)}(z),\qquad
k=n+1,\dots,N.$

Here

$\mathcal{N}_{jk}^{(1)}(z)=(-1)^{N-1}\hat{\lambda}_{1}(u_{j})h(\bar{v},u_{j})\left(\frac{h(\bar{\eta},u_{j})}{h(v_{k},u_{j})}-z\frac{g(v_{k},u_{j})}{g(\bar{\eta},u_{j})}\right)\\\
+\hat{\lambda}_{2}(u_{j})h(u_{j},\bar{v})\left(\frac{g(u_{j},v_{k})}{g(u_{j},\bar{\eta})}-z\frac{h(u_{j},\bar{\eta})}{h(u_{j},v_{k})}\right),\qquad
k=1,\dots,n,$ (5.17)

and

$\mathcal{N}_{jk}^{(2)}(z)=(-1)^{N}\hat{\lambda}_{1}(u_{j})h(\bar{v},u_{j})\left(\frac{z}{g(\bar{\eta}_{k},u_{j})}-h(\bar{\eta}_{k},u_{j})\right)\\\
+\hat{\lambda}_{2}(u_{j})h(u_{j},\bar{v})\left(\frac{1}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right),\qquad
k=n+1,\dots,N.$ (5.18)

Now we can transform $\det_{N}\mathcal{N}_{jk}(z)$ using corollary A.1 (see,
in particular, (A.29)). Thus,

$\det_{N}\mathcal{N}(z)=(z-1)^{N-n}\det_{N}\widetilde{\mathcal{N}}(z).$ (5.19)

Here $\widetilde{\mathcal{N}}_{jk}(z)=\mathcal{N}^{(1)}_{jk}(z)$ for
$k=1,\dots,n$, and

$\widetilde{\mathcal{N}}_{jk}(z)=(-1)^{n+1}\frac{\hat{\lambda}_{1}(u_{j})h(\bar{v},u_{j})}{g(u_{j},\bar{\eta}_{k})}-\hat{\lambda}_{2}(u_{j})h(u_{j},\bar{v})h(u_{j},\bar{\eta}_{k}),\qquad
k=n+1,\dots,N.$ (5.20)

Substituting this result into (5.15) we obtain

$S^{n,N}(\bar{v},\bar{u})=(-\mu)^{N}\left(\frac{\kappa^{-}}{\rho_{1}}\right)^{n}\lambda_{2}(\bar{v})\Delta^{\prime}(\bar{v})\Delta^{\prime}(\bar{\eta})g(\bar{v},\bar{\eta})\Delta(\bar{u})\det_{N}\widetilde{\mathcal{N}}_{jk}(1).$
(5.21)

In this representation, the parameters
$\bar{\eta}=\\{\eta_{n+1},\dots,\eta_{N}\\}$ remain arbitrary complex numbers.

###  5.4 Particular case

We did not impose any restrictions on the parameters $\bar{u}$ in the
representation (5.21). Therefore, the vector $|\Phi^{(2)}(\bar{u})\rangle$,
generally speaking, is off-shell. Consider a particular case $\alpha=1$. Then
the dual vector $\langle\Phi^{(1)}(\bar{v})|$ is an eigenvector of the
ordinary transfer matrix $t_{11}(z)+t_{22}(z)$. On the other hand, we find
$\rho_{1}=-\rho_{2}$ from the condition $\alpha=-\rho_{2}/\rho_{1}$. Then the
inhomogeneous term vanishes in the modified Bethe equations (3.17). We obtain

$\hat{\lambda}_{1}(u_{j})=(-1)^{N}\frac{\kappa+\rho}{\tilde{\kappa}-\rho}\;\frac{h(u_{j},\bar{u})}{h(\bar{u},u_{j})}\,\hat{\lambda}_{2}(u_{j}),\qquad
j=1,\dots,N,$ (5.22)

where $\rho=\rho_{1}=-\rho_{2}$. At the same time, the Bethe vectors
$|\Phi^{(2)}(\bar{u})\rangle$ remain modified in this particular case. They
are still given by (3.16).

We now require the vector $|\Phi^{(2)}(\bar{u})\rangle$ to be an on-shell
modified Bethe vector. For this, it is sufficient to substitute conditions
(5.22) into representation (5.21). After simple algebra, we obtain

$S^{n,N}(\bar{v},\bar{u})=(-\mu)^{N}\left(\frac{\rho}{\kappa^{-}}\right)^{N-n}\lambda_{2}(\bar{v})\lambda_{2}(\bar{u})h(\bar{u},\bar{v})\Delta^{\prime}(\bar{v})\Delta^{\prime}(\bar{\eta})g(\bar{v},\bar{\eta})\Delta(\bar{u})\det_{N}\Omega_{jk}.$
(5.23)

Here the matrix $\Omega$ consists of two parts:

$\Omega_{jk}=\frac{g(u_{j},v_{k})}{g(u_{j},\bar{\eta})}-\frac{h(u_{j},\bar{\eta})}{h(u_{j},v_{k})}-V_{j}\left(\frac{h(\bar{\eta},u_{j})}{h(v_{k},u_{j})}-\frac{g(v_{k},u_{j})}{g(\bar{\eta},u_{j})}\right),\qquad
k=1,\dots,n,$ (5.24)

and

$\Omega_{jk}=(-1)^{N+n+1}\frac{V_{j}}{g(u_{j},\bar{\eta}_{k})}-h(u_{j},\bar{\eta}_{k}),\qquad
k=n+1,\dots,N,$ (5.25)

where

$V_{j}=\frac{\kappa+\rho}{\tilde{\kappa}-\rho}\;\frac{h(u_{j},\bar{u})h(\bar{v},u_{j})}{h(\bar{u},u_{j})h(u_{j},\bar{v})}.$
(5.26)

It is easy to see that in a particular case $n=N$, we reproduce a determinant
representation for the scalar product of the usual on-shell and the twisted
on-shell Bethe vectors.

## Conclusion

We have considered overlaps of Bethe vectors with a diagonal twist and
modified Bethe vectors. We have shown that under one additional condition on
the twist parameters, such an overlap has a determinant representation. At the
same time, the modified Bethe vector can remain an off-shell vector. In this
sense, the result obtained is analogous to the scalar product of on-shell and
off-shell Bethe vectors [13].

However, despite this similarity, the resulting determinant representation has
an entirely new structure. Namely, it contains a set of arbitrary parameters.
This representation was obtained thanks to new formulas for the MID. These new
formulas contain a set of arbitrary complex parameters. Depending on the
specific task, these parameters can be chosen in the most convenient way. Such
formulas can find applications in models with higher symmetry rank,
particularly for $GL(3)$-invariant spin chains [17].

The resulting determinant representation for the overlap remains valid if the
modified Bethe vector is the eigenvector of the modified transfer matrix. In
this case, the parameters of the modified vector satisfy the inhomogeneous
Bethe equations (3.17). We have considered only one specific case, when the
inhomogeneous term in the Bethe equations disappears. The more general case
requires further study.

One of the apparent directions for further research is the application of the
obtained results to nonequilibrium physics. It is related to calculating the
density of states and the thermodynamic limit. Another area to consider is
models on a segment and overlaps under diagonal and non-diagonal boundary
conditions. In particular, it would be desirable to obtain a modified
Tsuchiya’s determinant [18]. It is quite possible that the method of reducing
overlaps to a system of linear equations [19] gives simple access to new
formulas.

## Acknowledgements

The work was performed at the Steklov Mathematical Institute of Russian
Academy of Sciences, Moscow. This work is supported by the Russian Science
Foundation under grant 19-11-00062.

##  Appendix A Properties of the MID

###  A.1 Transformations of the MID

In this appendix, we list several properties of the MID. A more detailed list
together with the proofs can be found in [7].

The MID and the conjugated one are related by

$\overline{K}_{n,m}^{(z)}(\bar{u}|\bar{v})=(1-z)^{m-n}K_{m,n}^{(z)}(\bar{v}|\bar{u}).$
(A.1)

The MID possesses the following property under the shift of one of the set of
parameters:

$K_{n,m}^{(z)}(\bar{u}-c|\bar{v})=\frac{(-z)^{n}(1-z)^{m-n}}{f(\bar{v},\bar{u})}K_{m,n}^{(1/z)}(\bar{v}|\bar{u}).$
(A.2)
$\overline{K}_{n,m}^{(z)}(\bar{u}+c|\bar{v})=\frac{(-z)^{n}(1-z)^{m-n}}{f(\bar{u},\bar{v})}\overline{K}_{m,n}^{(1/z)}(\bar{v}|\bar{u}).$
(A.3)

Some bilinear combinations of the MID reduce to a new MID.

######  Proposition A.1.

Let $\bar{\xi}$, $\bar{u}$, and $\bar{v}$ be sets of arbitrary complex numbers
such that $\\#\bar{\xi}=l$, $\\#\bar{u}=n$, and $\\#\bar{v}=m$. Then

$\sum_{\\{\bar{\xi}_{\scriptscriptstyle\rm I},\bar{\xi}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{\xi}}z_{2}^{l_{\scriptscriptstyle\rm
I}}K_{n,l_{\scriptscriptstyle\rm
I}}^{(z_{1})}(\bar{u}|\bar{\xi}_{\scriptscriptstyle\rm
I})K_{m,l_{\scriptscriptstyle\rm
II}}^{(z_{2})}(\bar{v}|\bar{\xi}_{\scriptscriptstyle\rm
II})f(\bar{\xi}_{\scriptscriptstyle\rm II},\bar{\xi}_{\scriptscriptstyle\rm
I})f(\bar{u},\bar{\xi}_{\scriptscriptstyle\rm
II})=K_{n+m,l}^{(z_{1}z_{2})}(\\{\bar{u},\bar{v}\\}|\bar{\xi}).$ (A.4)

Here $l_{\scriptscriptstyle\rm I}=\\#\bar{\xi}_{\scriptscriptstyle\rm I}$ and
$l_{\scriptscriptstyle\rm II}=\\#\bar{\xi}_{\scriptscriptstyle\rm II}$. The
sum is taken with respect to all partitions
$\\{\bar{\xi}_{\scriptscriptstyle\rm I},\bar{\xi}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{\xi}$. There is no any restriction on the cardinalities of
the subsets.

Replacing $K_{n,l_{\scriptscriptstyle\rm
I}}^{(z_{1})}(\bar{u}|\bar{\xi}_{\scriptscriptstyle\rm I})$ in (A.4) by the
conjugated MID via (A.1) and (A.2) we obtain

$\sum_{\\{\bar{\xi}_{\scriptscriptstyle\rm I},\bar{\xi}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{\xi}}\left(-\frac{z_{2}}{z_{1}}\right)^{l_{\scriptscriptstyle\rm
I}}\overline{K}_{n,l_{\scriptscriptstyle\rm
I}}^{(z_{1})}(\bar{u}|\bar{\xi}_{\scriptscriptstyle\rm
I})K_{m,l_{\scriptscriptstyle\rm
II}}^{(z_{2})}(\bar{v}|\bar{\xi}_{\scriptscriptstyle\rm
II})f(\bar{\xi}_{\scriptscriptstyle\rm II},\bar{\xi}_{\scriptscriptstyle\rm
I})=f(\bar{\xi},\bar{u})K_{n+m,l}^{(z_{1}z_{2})}(\\{\bar{u}-c,\bar{v}\\}|\bar{\xi}).$
(A.5)

Setting here $z_{1}=z_{2}=1$ we arrive at

$\sum_{\\{\bar{\xi}_{\scriptscriptstyle\rm I},\bar{\xi}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{\xi}}(-1)^{l_{\scriptscriptstyle\rm
I}}\overline{K}_{n,l_{\scriptscriptstyle\rm
I}}^{(1)}(\bar{u}|\bar{\xi}_{\scriptscriptstyle\rm
I})K_{m,l_{\scriptscriptstyle\rm
II}}^{(1)}(\bar{v}|\bar{\xi}_{\scriptscriptstyle\rm
II})f(\bar{\xi}_{\scriptscriptstyle\rm II},\bar{\xi}_{\scriptscriptstyle\rm
I})=f(\bar{\xi},\bar{u})K_{n+m,l}^{(1)}(\\{\bar{u}-c,\bar{v}\\}|\bar{\xi}).$
(A.6)

###  A.2 Other determinants related to the MID

######  Proposition A.2.

Let $\bar{u}=\\{u_{1},\dots,u_{N}\\}$,
$\bar{\eta}=\\{\eta_{1},\dots,\eta_{N}\\}$, and $z$ be arbitrary complex
numbers. Let $F_{k}^{(1)}(u)$ and $F_{k}^{(2)}(u)$, $k=1,\dots,N$, be two sets
of functions

$F_{k}^{(1)}(u)=\phi_{1}(u)\left(\frac{z}{g(\bar{\eta}_{k},u)}-h(\bar{\eta}_{k},u)\right)+\phi_{2}(u)\left(\frac{1}{g(u,\bar{\eta}_{k})}-zh(u,\bar{\eta}_{k})\right),$
(A.7)
$F_{k}^{(2)}(u)=\frac{(-1)^{N-1}\phi_{1}(u)}{g(u,\bar{\eta}_{k})}-\phi_{2}(u)h(u,\bar{\eta}_{k}).$
(A.8)

Here $\phi_{\ell}(z)$ ($\ell=1,2$) are two arbitrary functions. Let us compose
two $N\times N$ matrices $\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ with the entries

$\hat{F}_{jk}^{(1)}=F_{k}^{(1)}(u_{j}),\qquad\hat{F}_{jk}^{(2)}=F_{k}^{(2)}(u_{j}).$
(A.9)

Then

$\det_{N}\hat{F}^{(1)}=(z-1)^{N}\det_{N}\hat{F}^{(2)}.$ (A.10)

Proof. Obviously, both determinants can be presented in the form

$\det_{N}\hat{F}^{(\ell)}=\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}X^{(\ell)}(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})\phi_{1}(\bar{u}_{\scriptscriptstyle\rm
I})\phi_{2}(\bar{u}_{\scriptscriptstyle\rm II}),\qquad\ell=1,2,$ (A.11)

where the coefficients $X^{(\ell)}(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})$ do not depend on $\phi_{1}$ and
$\phi_{2}$. Since $\phi_{\ell}$ are arbitrary functions, equation (A.10) holds
if and only if

$X^{(1)}(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=(z-1)^{N}X^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})$ (A.12)

for an arbitrary partition $\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}$. However, it is easy to
see that without loss of generality, it is enough to prove (A.12) for
$\bar{u}_{\scriptscriptstyle\rm I}=\\{u_{1},\dots,u_{p}\\}$ and
$\bar{u}_{\scriptscriptstyle\rm II}=\\{u_{p+1},\dots,u_{N}\\}$. Here $p$ is an
arbitrary integer from the set $\\{0,1,\dots,N\\}$.

To obtain the coefficients $X^{(\ell)}(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})$, we should set $\phi_{2}(u_{j})=0$ for
$j=1,\dots,p$ and $\phi_{1}(u_{j})=0$ for $j=p+1,\dots,N$ in (A.7) and (A.8).
We obtain

$X^{(\ell)}(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=\det_{N}\Phi^{(\ell)},\qquad\ell=1,2,$ (A.13)

where

$\displaystyle\Phi^{(1)}_{jk}=\frac{z}{g(\bar{\eta}_{k},u_{j})}-h(\bar{\eta}_{k},u_{j}),\qquad
j=1,\dots,p,$ (A.14)
$\displaystyle\Phi^{(1)}_{jk}=\frac{1}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k}),\qquad
j=p+1,\dots,N,$

and

$\displaystyle\Phi^{(2)}_{jk}=\frac{(-1)^{m-1}}{g(u_{j},\bar{\eta}_{k})},\qquad
j=1,\dots,p,$ (A.15)
$\displaystyle\Phi^{(2)}_{jk}=-h(u_{j},\bar{\eta}_{k}),\qquad j=p+1,\dots,N.$

Let $u_{j}=u^{\prime}_{j}+c$ for $j=1,\dots,p$ and $u_{j}=u^{\prime}_{j}$ for
$j=p+1,\dots,N$. Using
$1/g(u_{j},\bar{\eta}_{k})=h(u^{\prime}_{j},\bar{\eta}_{k})$ and
$h(\bar{\eta}_{k},u_{j})=(-1)^{N-1}/g(u^{\prime}_{j},\bar{\eta}_{k})$ (see
(2.12)) for $k=1,\dots,p$, we obtain

$X^{(\ell)}(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=(-1)^{(\ell+p-1)N}\det_{N}\widetilde{\Phi}^{(\ell)},\qquad\ell=1,2,$
(A.16)

where

$\begin{aligned}
&\widetilde{\Phi}^{(1)}_{jk}=\frac{1}{g(u^{\prime}_{j},\bar{\eta}_{k})}-zh(u^{\prime}_{j},\bar{\eta}_{k}),\\\
&\widetilde{\Phi}^{(2)}_{jk}=h(u^{\prime}_{j},\bar{\eta}_{k}),\end{aligned}\qquad\qquad
j,k=1,\dots,N.$ (A.17)

It is easy to see that $\det_{N}\widetilde{\Phi}^{(2)}$ reduces to the Cauchy
determinant:

$\det_{N}\widetilde{\Phi}^{(2)}=h(\bar{u}^{\prime},\bar{\eta})\det_{N}\left(\frac{1}{h(u^{\prime}_{j},\eta_{k})}\right)=\frac{1}{\Delta(\bar{\eta})\Delta^{\prime}(\bar{u}^{\prime})}.$
(A.18)

The determinant of $\widetilde{\Phi}^{(1)}$ is computed in corollary 4.1. Due
to (4.13), we have

$\det_{N}\widetilde{\Phi}^{(1)}=\frac{(1-z)^{N}}{\Delta(\bar{\eta})\Delta^{\prime}(\bar{u}^{\prime})}.$
(A.19)

Comparing this equation with (A.18) and using (A.16) we see that

$X^{(1)}(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=(z-1)^{N}X^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}).$ (A.20)

∎

######  Corollary A.1.

Let $0\leq n\leq N$, and let $\bar{u}=\\{u_{1},\dots,u_{N}\\}$,
$\bar{\eta}=\\{\eta_{n+1},\dots,\eta_{N}\\}$, and $z$ be arbitrary complex
numbers ($\bar{\eta}=\emptyset$ for $n=N$). Let us compose two $N\times N$
matrices $\hat{F}^{(01)}$ and $\hat{F}^{(02)}$ with the entries

$\hat{F}_{jk}^{(01)}=\begin{cases}F_{k}^{(0)}(u_{j}),\qquad k=1,\dots,n,\\\
F_{k}^{(1)}(u_{j}),\qquad k=n+1,\dots,N,\end{cases}$ (A.21)
$\hat{F}_{jk}^{(02)}=\begin{cases}F_{k}^{(0)}(u_{j}),\qquad k=1,\dots,n,\\\
F_{k}^{(2)}(u_{j}),\qquad k=n+1,\dots,N.\end{cases}$ (A.22)

Here $F_{k}^{(1)}(u)$ and $F_{k}^{(2)}(u)$ respectively are given by (A.7) and
(A.8), while $F_{k}^{(0)}(u)$, $k=1,\dots,n$, are arbitrary functions. Then

$\det_{N}\hat{F}^{(01)}=(z-1)^{N-n}\det_{N}\hat{F}^{(02)}.$ (A.23)

Proof. Developing the determinant $\det\hat{F}^{(01)}$ with respect to the
first $k$ columns we obtain

$\det_{N}\hat{F}^{(01)}=\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}(-1)^{P_{\scriptscriptstyle\rm I,\scriptscriptstyle\rm
II}}\det_{n}\Big{(}F_{k}^{(0)}(u^{\scriptscriptstyle\rm
I}_{j})\Big{)}\det_{N-n}\Big{(}F_{k}^{(1)}(u^{\scriptscriptstyle\rm
II}_{j})\Big{)}.$ (A.24)

Here the sum is taken over partitions $\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}$ such that
$\\#\bar{u}_{\scriptscriptstyle\rm I}=n$ and
$\\#\bar{u}_{\scriptscriptstyle\rm II}=N-n$. Notation
$u^{\scriptscriptstyle\rm I}_{j}$ (resp. $u^{\scriptscriptstyle\rm II}_{j}$)
means the $j$th element of the subset $\bar{u}_{\scriptscriptstyle\rm I}$
(resp. $\bar{u}_{\scriptscriptstyle\rm II}$). Finally,
$P_{\scriptscriptstyle\rm I,\scriptscriptstyle\rm II}$ is the parity of the
partition $\\{\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}$.

Due to proposition A.2

$\det_{N-n}\Big{(}F_{k}^{(1)}(u^{\scriptscriptstyle\rm
II}_{j})\Big{)}=(z-1)^{N-n}\det_{N-n}\Big{(}F_{k}^{(2)}(u^{\scriptscriptstyle\rm
II}_{j})\Big{)},$ (A.25)

for arbitrary subset $\bar{u}_{\scriptscriptstyle\rm II}$. Hence,

$\det_{N}\hat{F}^{(01)}=(z-1)^{N-n}\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}(-1)^{P_{\scriptscriptstyle\rm I,\scriptscriptstyle\rm
II}}\det_{n}\Big{(}F_{k}^{(0)}(u^{\scriptscriptstyle\rm
I}_{j})\Big{)}\det_{N-n}\Big{(}F_{k}^{(2)}(u^{\scriptscriptstyle\rm
II}_{j})\Big{)}.$ (A.26)

Taking the sum over partitions we arrive at (A.23).∎

In particular, setting

$\phi_{1}(u)=(-1)^{N}\hat{\lambda}_{1}(u)h(\bar{v},u),\qquad\phi_{2}(u)=\hat{\lambda}_{2}(u)h(u,\bar{v}),$
(A.27)

we obtain

$F_{k}^{(1)}(u_{j})=\mathcal{N}^{(2)}_{jk},\qquad k=n+1,\dots,N,$ (A.28)

where $\mathcal{N}^{(2)}_{jk}$ is given by (5.18). Due to corollary A.1, we
obtain

$\det_{N}\hat{F}^{(01)}=(z-1)^{N-n}\det_{N}\hat{F}^{(02)}.$ (A.29)

Here

$\hat{F}_{jk}^{(01)}=\begin{cases}F_{k}^{(0)}(u_{j}),\qquad k=1,\dots,n,\\\
\mathcal{N}^{(2)}_{jk},\qquad k=n+1,\dots,N,\end{cases}$ (A.30)
$\hat{F}_{jk}^{(02)}=\begin{cases}F_{k}^{(0)}(u_{j}),\qquad k=1,\dots,n,\\\
(-1)^{n+1}\hat{\lambda}_{1}(u_{j})\frac{h(\bar{v},u_{j})}{g(u_{j},\bar{\eta}_{k})}-\hat{\lambda}_{2}(u_{j})h(u_{j},\bar{v})h(u_{j},\bar{\eta}_{k}),\qquad
k=n+1,\dots,N.\end{cases}$ (A.31)

and $F_{k}^{(0)}(u)$, $k=1,\dots,n$, are arbitrary functions.

###  A.3 Summation formula

Let a function $H(\bar{u})$ of $N$ variables $\bar{u}=\\{u_{1},\dots,u_{N}\\}$
be defined as follows:

$H(\bar{u})=\Delta(\bar{u})\det_{N}\Phi_{k}(u_{j}).$ (A.32)

Here $\Phi_{k}(u)$ is a set of one-variable functions.

######  Proposition A.3.

Let $\phi_{1}(u)$ and $\phi_{2}(u)$ be one-variable functions. Then

$\sum_{\\{\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}\phi_{1}(\bar{u}_{\scriptscriptstyle\rm
I})\phi_{2}(\bar{u}_{\scriptscriptstyle\rm
II})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})H(\\{\bar{u}_{\scriptscriptstyle\rm I}-c,\bar{u}_{\scriptscriptstyle\rm
II}\\})=\Delta(\bar{u})\det_{N}\Bigl{(}\phi_{1}(u_{j})\Phi_{k}(u_{j}-c)+\phi_{2}(u_{j})\Phi_{k}(u_{j})\Bigr{)}.$
(A.33)

The sum is taken over all possible partitions
$\\{\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}$, and we used the shorthand notation for the products of
the functions $\phi_{\ell}(u)$, $\ell=1,2$.

Proof. The proof is similar to the one of proposition A.2. The rhs of (A.33)
can be presented as follows:

$\Delta(\bar{u})\det_{N}\Bigl{(}\phi_{1}(u_{j})\Phi_{k}(u_{j}-c)+\phi_{2}(u_{j})\Phi_{k}(u_{j})\Bigr{)}=\sum_{\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}X(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})\phi_{1}(\bar{u}_{\scriptscriptstyle\rm
I})\phi_{2}(\bar{u}_{\scriptscriptstyle\rm II}),$ (A.34)

where the coefficients $X(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})$ do not depend on $\phi_{1}$ and
$\phi_{2}$. Since $\phi_{\ell}$ are arbitrary functions, equation (A.33) holds
if and only if

$X(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})H(\\{\bar{u}_{\scriptscriptstyle\rm I}-c,\bar{u}_{\scriptscriptstyle\rm
II}\\}),$ (A.35)

for an arbitrary partition $\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\vdash\bar{u}$. Due to the symmetry of
(A.33) over $\bar{u}$, it is enough to prove (A.35) for
$\bar{u}_{\scriptscriptstyle\rm I}=\\{u_{1},\dots,u_{p}\\}$ and
$\bar{u}_{\scriptscriptstyle\rm II}=\\{u_{p+1},\dots,u_{N}\\}$. Here $p$ is an
arbitrary integer from the set $\\{0,1,\dots,N\\}$.

To obtain the coefficients $X(\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II})$, we should set $\phi_{2}(u_{j})=0$ for
$j=1,\dots,p$ and $\phi_{1}(u_{j})=0$ for $j=p+1,\dots,N$ in (A.33). We obtain

$X(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=\Delta(\bar{u}_{\scriptscriptstyle\rm
I})\Delta(\bar{u}_{\scriptscriptstyle\rm II})g(\bar{u}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm I})\det_{N}\widetilde{\Phi},$ (A.36)

where

$\displaystyle\widetilde{\Phi}_{jk}=\Phi_{k}(u_{j}-c),\qquad j=1,\dots,p,$
(A.37) $\displaystyle\widetilde{\Phi}_{jk}=\Phi_{k}(u_{j}),\qquad
j=p+1,\dots,N.$

Let $u_{j}=u^{\prime}_{j}+c$ for $j=1,\dots,p$ and $u_{j}=u^{\prime}_{j}$ for
$j=p+1,\dots,N$. Using $g(x,y+c)=-1/h(y,x)$ for any $x$ and $y$, we obtain

$\Delta(\bar{u}_{\scriptscriptstyle\rm
I})\Delta(\bar{u}_{\scriptscriptstyle\rm II})g(\bar{u}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm
I})=(-1)^{p(N-p)}\frac{\Delta(\bar{u}^{\prime}_{\scriptscriptstyle\rm
I})\Delta(\bar{u}^{\prime}_{\scriptscriptstyle\rm
II})}{h(\bar{u}^{\prime}_{\scriptscriptstyle\rm
I},\bar{u}^{\prime}_{\scriptscriptstyle\rm
II})}=\frac{\Delta(\bar{u}^{\prime})}{f(\bar{u}^{\prime}_{\scriptscriptstyle\rm
I},\bar{u}^{\prime}_{\scriptscriptstyle\rm II})}.$ (A.38)

Then (A.36) takes the form

$X(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})=\frac{\Delta(\bar{u}^{\prime})}{f(\bar{u}^{\prime}_{\scriptscriptstyle\rm
I},\bar{u}^{\prime}_{\scriptscriptstyle\rm
II})}\det_{N}\Phi_{k}(u^{\prime}_{j})=\frac{H(\bar{u}^{\prime})}{f(\bar{u}^{\prime}_{\scriptscriptstyle\rm
I},\bar{u}^{\prime}_{\scriptscriptstyle\rm II})}.$ (A.39)

Turning back to the original variables and using $f(x-c,y)=1/f(y,x)$ for any
$x$ and $y$ we arrive at (A.35). ∎

We deal with a particular case of the sum (A.33) in equation (5.10). Indeed,
let

$\phi_{1}(u)=\lambda_{1}(u)f(\bar{v},u),\qquad\phi_{2}(u)=\frac{\rho_{2}}{\rho_{1}}\lambda_{2}(u),\qquad\Phi_{k}(u)=\frac{f(u,\bar{v})}{g(u,\bar{\eta}_{k})}-zh(u,\bar{\eta}_{k}),$
(A.40)

where $\bar{\eta}=\\{\eta_{1},\dots,\eta_{N}\\}$ are arbitrary complex
numbers. Then

$\Delta^{\prime}(\bar{\eta})H(\bar{u})=(1-z)^{N-n}K^{(z)}_{N,n}(\bar{u}|\bar{v}).$
(A.41)

Hence, due to proposition A.3, we obtain

$\sum_{\\{\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II}\\}\vdash\bar{u}}\left(\frac{\rho_{2}}{\rho_{1}}\right)^{N_{\scriptscriptstyle\rm
II}}\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{1}(\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{v},\bar{u}_{\scriptscriptstyle\rm
I})K^{(1)}_{N,n}(\\{\bar{u}_{\scriptscriptstyle\rm
I}-c,\bar{u}_{\scriptscriptstyle\rm II}\\}|\bar{v})\\\
=(1-z)^{N-n}\Delta^{\prime}(\bar{\eta})\Delta(\bar{u})\det_{N}\mathcal{M}_{jk},$
(A.42)

where

$\mathcal{M}_{jk}=\lambda_{1}(u_{j})f(\bar{v},u_{j})\left(\frac{f(u_{j}-c,\bar{v})}{g(u_{j}-c,\bar{\eta}_{k})}-zh(u_{j}-c,\bar{\eta}_{k})\right)\\\
+\frac{\rho_{2}}{\rho_{1}}\lambda_{2}(u_{j})\left(\frac{f(u_{j},\bar{v})}{g(u_{j},\bar{\eta}_{k})}-zh(u_{j},\bar{\eta}_{k})\right).$
(A.43)

Using equations (2.12) we easily transform this result to representation
(5.12).

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