# On a Relation among Bi-orthogonal system, Quadratic Non-Hermitian Boson
operators with real spectrum and Partial $\mathcal{PT}$ symmetry in Fock Space

Arindam Chakraborty11footnotemark: 1 Department of Physics, Heritage Institute
of Technology, Kolkata-700107, India  [<EMAIL_ADDRESS>
###### Abstract

A new kind of symmetry called partial $\mathcal{PT}$ symmetry (henceforth
$\partial_{\mathcal{PT}}$) has been considered for non-hermitian quadratic
boson operators obtained from a bi-orthogonal set of vectors in
$\mathbf{C}^{2}$. The symmetry behaviour has been understood in Fock space
considered as a Reproducing Kernel Hilbert Space(RKHS). The reality of
eigenvalues and its connection to the possibility of the aforesaid symmetry
(and symmetry breaking) are studied in terms of a deformation parameter
responsible for non-hermiticity.

###### keywords:

Non-hermitian Quantum Mechanics, Partial $PT$-symmetry, bi-orthogonal system,
Fock Space, Reproducing Kernel Hilbert Space.

††journal: Journal of LaTeX Templates

url]www.elsevier.com

##  1 Introduction

In recent years the use of non-hermitian operators with real spectrum in the
context of space-time reflection symmetry (${\mathcal{PT}}$-symmetry) has
gained much relevance in modern quantum methods [1, 2, 3, 4, 5, 6, 7, 8]. The
issue of ${\mathcal{PT}}$-symmetry in non-hermitian Hamiltonian [9] has been
widely studied in open regimes with balanced gain and loss like laser
absorbers [10], ultra-cold threshold phonon lasers [11], defect states in
special beam dynamics in spatial lattices [12],quantum circuit based on
nuclear magnetic resonance [13], microwave cavities [14], super-conductivity
[15], Bose-Einstein condensates [16, 17], optical wave guides [18] and quantum
entanglement[19]. The theoretical developments in terms of space and algebraic
structures of non-hermitian systems [20, 21, 22], the role of
${\mathcal{PT}}$-symmetric operators and their significance in pseudo-
hermitian quantum mechanics [23] have also become some interesting areas of
investigation [24, 25, 26].

The extension of symmetry argument for systems involving two or more particles
has been attempted earlier[16, 27]. However, the reality of the spectrum does
not necessarily correspond to the existence of global
${\mathcal{PT}}$-symmetry. In fact in a recent article by Beygi et. al. [28] a
variable dependent $\mathcal{PT}$ symmetry behaviour has been investigated in
a model of coupled oscillator with purely imaginary coupling term along with
the possibilities of real and imaginary eigenvalues. Our present objective is
to elaborate the symmetry argument in a Fock Space setting with the help of
weighted composition conjugation [29, 30, 31] representing complex symmetries
[32, 33] and acting on a Reproducing Kernel Hilbert Space[34, 35]. The
following discussion also shows that a spectrum generating algebra can be
constructed starting from a bi-orthogonal system of vectors in
$\mathbf{C}^{2}$ and reality of the spectrum has a direct relation with bi-
orthogonality as well as with the existence and breaking of
$\partial_{\mathcal{PT}}$-symmetry.

##  2 Quadratic Boson Operator and $\partial_{\mathcal{PT}}$ symmetry

We consider a specific quadratic boson operator
$B=\frac{1}{2}\sum_{m,n=1}^{3}B_{mn}a^{\dagger}_{m}a_{n}$ where,
$\\{a_{\alpha},a^{\dagger}_{\alpha}|\alpha=1,2\\}$ are boson operators with
$[a_{\alpha},a^{\dagger}_{\beta}]-\delta_{\alpha\beta}=[a_{\alpha},a_{\beta}]=[a^{\dagger}_{\alpha},a^{\dagger}_{\beta}]=0$
and $B_{11}=c_{1},B_{22}=c_{2},B_{12}=B_{21}=i\alpha$, $c_{1}$, $c_{2}$ and
$\alpha$ being real. We consider $(\mathcal{F}^{2}(\mathbf{C}^{2}))$ as the
Fock (or Segal-Bargmann) space which is a separable complex Hilbert space of
entire functions (of the complex variables $\zeta_{1}$ and $\zeta_{2}$)
equipped with an inner-product

$\displaystyle\langle\psi,\phi\rangle=\int_{\mu(\zeta_{1})}\int_{\mu(\zeta_{2})}\psi(\zeta_{1},\zeta_{2})\overline{\phi(\zeta_{1},\zeta_{2})}$
$\displaystyle\;\;{\rm{with}}\int_{\mu(\zeta_{1})}\int_{\mu(\zeta_{2})}\equiv\int\int
d{\mu(\zeta_{1})}d{\mu(\zeta_{2})}$

Here, ${d}{\mu(z)}=\frac{1}{\pi}e^{-|z|^{2}}{d}({\rm{Re}}(z))d({\rm{Im}}(z))$
represents the relevant Gaussian measure relative to the complex variable $z$.

###### Definition 1

In a Fock space of one complex variable $\zeta$, the weighted composition
conjugation [29] can be given by the expression

$\displaystyle\mathcal{W}_{(\vartheta,\eta,\upsilon)}\psi(\zeta)=\upsilon
e^{\eta\zeta}\overline{\psi(\overline{\vartheta\zeta+\eta})}.$ (1)

Here, $\zeta$ is a complex variable and $\\{\vartheta,\eta,\upsilon\\}$ are
complex numbers satisfying the set of necessary and sufficient conditions :
$|\vartheta|=1,\bar{\vartheta}\eta+\bar{\eta}=0$ and
$|\upsilon|^{2}e^{|\eta|^{2}}=1$. The anti-linear operator
$\mathcal{W}_{(\vartheta,\eta,\upsilon)}$ is involutive and isometric and
hence a conjugation. The action of the operator $\mathcal{PT}$ is equivalent
to the choice : $\vartheta=-1=-\upsilon,\eta=0$ which gives
$\mathcal{W}_{(\vartheta,0,1)}|_{\vartheta=-1}\psi(\zeta)=\overline{\psi(\overline{-\zeta})}.$
Similarly, the action of $\mathcal{T}$ involves the choice :
$\vartheta=1,\eta=0,\upsilon=1$ giving
$\mathcal{W}_{(\vartheta,0,1)}|_{\vartheta=1}\psi(\zeta)=\overline{\psi(\overline{\zeta})}$.

Considering $\psi\dot{=}\psi(\zeta_{1},\zeta_{2},\cdots\zeta_{n})$ as a
function of several complex variables one can define an operator
$\mathcal{W}_{(\vartheta_{j},\eta_{j},\upsilon_{j}:j=1\cdots n)}$ with the
action

$\displaystyle\mathcal{W}_{(\vartheta_{j},\eta_{j}=0,\upsilon_{j}=1:j=1\cdots
n)}\psi(\zeta_{1},\cdots,\zeta_{j},\cdots,\zeta_{n})=\overline{\psi(\overline{\vartheta_{1}\zeta_{1}},\cdots,\overline{\vartheta_{j}\zeta_{j}},\cdots,\overline{\vartheta_{n}\zeta_{n}})}.$
(2)

###### Definition 2

An operator
$\mathcal{W}^{(j)}_{n}=\mathcal{W}_{(\vartheta_{j},\eta_{j}=0,\upsilon_{j}=1;j=1\cdots
n)}|_{\vartheta_{1}=1,\cdots,\vartheta_{j}=-1,\cdots,\vartheta_{n}=1}$ is
called the $j$-th partial $\mathcal{PT}$ symmetry ($\partial_{\mathcal{PT}}$)
operator if

$\displaystyle\mathcal{W}^{(j)}_{n}\psi(\zeta_{1},\cdots,\zeta_{j},\cdots,\zeta_{n})=\overline{\psi(\bar{\zeta_{1}},\cdots,\overline{-\zeta_{j}},\cdots,\bar{\zeta_{n}})}.$
(3)

and an operator $\mathcal{W}_{n}$ is called a global $\mathcal{PT}$ symmetry
operator if

$\displaystyle\mathcal{W}_{n}\psi(\zeta_{1},\cdots,\zeta_{j},\cdots,\zeta_{n})=\overline{\psi(\overline{-\zeta_{1}},\cdots,\overline{-\zeta_{j}},\cdots,\overline{-\zeta_{n}})}.$
(4)

For our present purpose we shall only consider the operators $\mathcal{W}_{2}$
and $\\{\mathcal{W}_{2}^{(j)}:j=1,2\\}$. Now, global and partial
$\mathcal{PT}$ symmetries of any function $\psi(\zeta_{1},\zeta_{2})$ are
expressed through the relations
$\mathcal{W}_{2}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})$ and
$\mathcal{W}_{2}^{(j)}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})\>\>\forall\>\>j=1,2$
respectively.

For the symmetry behaviour of $B(c_{1},c_{2},i\alpha,i\alpha)$ we consider the
following notion of Reproducing Kernel Hilbert Space (RKHS).

###### Definition 3

A function of the form
$\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2})=z_{1}^{m_{1}}z_{2}^{m_{2}}e^{z_{1}\overline{\zeta_{1}}+z_{2}\overline{\zeta_{2}}}$
with $m_{1},m_{2}\in\mathbf{N}$ and
$\zeta_{j},z_{j}\in\mathbf{C}\>\>\forall\>\>j=1,2$ is called a kernel function
(or a Reproducing Kernel) which satisfies the condition

$\displaystyle\psi^{(m_{1},m_{2})}(\zeta_{1},\zeta_{2})=\left\langle\psi,\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\right\rangle=\int_{\mu(z_{1})}\int_{\mu(z_{2})}\psi(z_{1},z_{2})\overline{\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}}.$
(5)

Here, $\psi\in\mathcal{F}^{2}(\mathbf{C}^{2})$,
$\psi^{(m_{1},m_{2})}(\zeta_{1},\zeta_{2})=\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}\psi$
and $\psi^{(0,0)}(\zeta_{1},\zeta_{2})\equiv\psi(\zeta_{1},\zeta_{2})$. Hence
the

###### Proposition 1

$B(c_{1},c_{2},i\alpha,i\alpha)^{\star}\neq B(c_{1},c_{2},i\alpha,i\alpha)$
where for any operator $H$, $H^{\star}$ is defined as $\left\langle
H\psi(u_{1},u_{2}),\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\right\rangle=\left\langle\psi(u_{1},u_{2}),H^{\star}\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\right\rangle$.

The self-adjointness of the operator $B$ we can write the

###### Proposition 2

$B(c_{1},c_{2},i\alpha,i\alpha)$ is $\mathcal{W}_{2}$-self-adjoint i. e.;
$\mathcal{W}_{2}B(c_{1},c_{2},i\alpha,i\alpha)^{\star}\mathcal{W}_{2}=B(c_{1},c_{2},i\alpha,i\alpha)$
but it is not $\mathcal{W}_{2}^{(j)}$-self-adjoint.

Proof : We shall first show the case with $z_{1}\partial_{z_{1}}$. Considering
the conjugation operator $\mathcal{W}_{2}$ we find

$\displaystyle\mathcal{W}_{2}(z_{1}\partial_{z_{1}})^{\star}\mathcal{W}_{2}\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2})$
(6) $\displaystyle=$
$\displaystyle\mathcal{W}_{2}z_{1}\partial_{z_{1}}\mathcal{W}_{2}z_{1}^{m_{1}}z_{2}^{m_{2}}e^{z_{1}\overline{\zeta_{1}}+z_{2}\overline{\zeta_{2}}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{2}z_{1}\partial_{z_{1}}\overline{\overline{(-z_{1})}^{m_{1}}}\overline{\overline{(-z_{2})}^{m_{2}}}\overline{e^{\overline{-z_{1}}\overline{\zeta_{1}}+\overline{-z_{2}}\overline{\zeta_{2}}}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{2}z_{1}\partial_{z_{1}}(-1)^{m_{1}+m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}e^{-z_{1}{\zeta_{1}}-z_{2}{\zeta_{2}}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{2}z_{1}(-1)^{m_{1}+m_{2}}[z_{1}^{m_{1}}z_{2}^{m_{2}}(-{\zeta_{1}})e^{-z_{1}{\zeta_{1}}-z_{2}{\zeta_{2}}}+m_{1}z_{1}^{m_{1}-1}z_{2}^{m_{2}}e^{-z_{1}{\zeta_{1}}-z_{2}{\zeta_{2}}}]$
$\displaystyle=$ $\displaystyle-
z_{1}(-1)^{m_{1}+m_{2}}[(-1)^{m_{1}+m_{2}}z_{1}^{m_{1}}z_{2}^{m_{2}}(-\overline{\zeta_{1}})$
$\displaystyle+m_{1}(-1)^{m_{1}+m_{2}-1}z_{1}^{m_{1}-1}z_{2}^{m_{2}}]e^{z_{1}\overline{\zeta_{1}}+z_{2}\overline{\zeta_{2}}}$
$\displaystyle=$ $\displaystyle
z_{1}\partial_{z_{1}}\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2}).$

Similarly one can prove

$\mathcal{W}_{2}(iz_{1}\partial_{z_{2}})^{\star}\mathcal{W}_{2}\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2})=(iz_{2}\partial_{z_{1}})\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2})$

and

$\mathcal{W}_{2}(iz_{2}\partial_{z_{1}})^{\star}\mathcal{W}_{2}\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2})=(iz_{1}\partial_{z_{2}})\kappa^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(z_{1},z_{2}).$

Using these results in the expression of the $B(c_{1},c_{2},i\alpha,i\alpha)$,
the above proposition is verified. Similar calculation may be repeated to show
the absence of $\mathcal{W}_{2}^{(j)}$-self-adjointness.

Similar argument justifies the following proposition

###### Proposition 3

$B(c_{1},c_{2},i\alpha,i\alpha)$ has $\partial_{\mathcal{PT}}$ symmetry i. e.;
$\mathcal{W}_{2}^{(j)}B(c_{1},c_{2},i\alpha,i\alpha)$
$\mathcal{W}_{2}^{(j)}=B(c_{1},c_{2},i\alpha,i\alpha)$, for $j=1,2$ but it
lacks global $\mathcal{PT}$ symmetry i. e.;
$\mathcal{W}_{2}B(c_{1},c_{2},i\alpha,i\alpha)$ $\mathcal{W}_{2}\neq
B(c_{1},c_{2},i\alpha,i\alpha)$.

##  3 $\partial_{\mathcal{PT}}$ symmetry of the eigenstates of
$B(c_{1},c_{2},i\alpha,i\alpha)$

First we shall show that the relation between the reality of the eigenvalues
and $\partial_{\mathcal{PT}}$ symmetry of the eigen states of the Hamiltonian.
Since that the present operator leaves the homogeneous polynomial space of two
indeterminates $(\zeta_{1},\zeta_{2})$ invariant, let us consider the degree
of homogeneity $m$ and polynomial bases
$\\{f_{k}=\zeta_{1}^{m-k}\zeta_{2}^{k}:k=0\cdots m\\}$. The operator
$B(c_{1},c_{2},i\alpha,i\alpha)$ takes the following tridiagonal
representation

$\displaystyle B=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccccccc}\beta_{m.0}&i\alpha&0&0&0&\cdots&0&0\\\
im\alpha&\beta_{m,1}&2i\alpha&0&0&\cdots&0&0\\\
0&i\alpha(m-1)&\beta_{m,2}&3i\alpha&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&\beta_{m,m-1}&im\alpha\\\
0&0&0&0&0&\cdots&i\alpha&\beta_{m,m}\end{array}\right).$

where, $\\{\beta_{m,k}=c_{1}(m-k)+c_{2}k:k=0,\dots,m\\}$ The the eigenvalues
of such a matrix can be found out with the help of the following theorem [36].

###### Theorem 1

Given a tri-diagonal matrix

${\mathcal{M}}=\left(\begin{array}[]{cccccccc}b_{0}&d_{0}&0&0&0&\cdots&0&0\\\
c_{0}&b_{1}&d_{1}&0&0&\cdots&0&0\\\ 0&c_{1}&b_{2}&d_{2}&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&b_{l-2}&d_{l-2}\\\
0&0&0&0&0&\cdots&c_{l-2}&b_{l-1}\end{array}\right)$ (8)

with $d_{i}\neq 0\>\>\forall\>\>i$, let us consider a polynomial
$Q_{n}(\lambda)$ that follows the well-known three term recursion relation
[37, 38]

$Q_{n+1}(\lambda)=\frac{1}{d_{n}}[(\lambda-
b_{n})Q_{n}(x)-c_{n-1}Q_{n-1}(\lambda)].$ (9)

If $Q_{-1}(\lambda)=0$ and $Q_{0}(\lambda)=1$ the eigenvalues are given by the
zeros of the polynomial $Q_{l}(\lambda)$ and eigenvector corresponding to
$j$-th eigenvalue $\lambda_{j}$ is given by the vector
$\left(Q_{0}(\lambda_{j})\>\>Q_{1}(\lambda_{j})\>\>\dots\>\>Q_{l-2}(\lambda_{j})\>\>Q_{l-1}(\lambda_{j})\right)^{\dagger}$

Concerning the symmetry of the eigenfunctions follows the

###### Corollary 1

If an eigenvalue of $B(c_{1},c_{2},i\alpha,i\alpha)$ is real the corresponding
eigenfunction in the homogeneous polynomial space is $\partial_{\mathcal{PT}}$
symmetric (or anti-symmetric). If an eigenvalue has non-zero imaginary part
the said symmetry is broken.

Proof : For real values of $c_{1},c_{2}$ and $\alpha$ the set of polynomials
${Q_{n}(\lambda)}$ becomes alternatively real and purely imaginary for any
real value of $\lambda$. The eigenfunction corresponding to $\lambda$ in any
polynomial space with degree of homogeneity $m$ can be given by
$\psi_{2s-1}(\zeta_{1},\zeta_{2})=A_{0}\zeta_{1}^{2s-1}+iA_{1}\zeta_{1}^{2s-2}\zeta_{2}+A_{2}\zeta_{1}^{2s-3}\zeta_{2}^{2}+\dots+iA_{2s-1}\zeta_{2}^{2s-1}\>\>{\rm
for}\>\>m=2s-1$ and
$\psi_{2s}(\zeta_{1},\zeta_{2})=B_{0}\zeta_{1}^{2s}+iB_{1}\zeta_{1}^{2s-1}\zeta_{2}+B_{2}\zeta_{1}^{2s-2}\zeta_{2}^{2}+\dots+B_{2s}\zeta_{2}^{2s}\>\>{\rm
for}\>\>m=2s$. Here, $\\{A_{l}\\}$ and $\\{B_{l}\\}$ are real functions for
real values of $\lambda$ with $A_{0}=B_{0}=1$. Now the action of
$\mathcal{W}_{2}^{(j)}$ on $\psi_{2s-1}$ is given by

$\displaystyle\mathcal{W}_{2}^{(1)}\psi_{2s-1}$ $\displaystyle=$
$\displaystyle\overline{A_{0}\overline{(-\zeta_{1})}^{2s-1}+iA_{1}\overline{(-\zeta_{1})}^{2s-2}\overline{(\zeta_{1})}+\dots+A_{2s-1}\overline{(\zeta_{2})}^{2s-1}}$
$\displaystyle=-\psi_{2s-1}$ (10)

Similarly, $\mathcal{W}_{2}^{(2)}\psi_{2s-1}=\psi_{2s-1}$ and
$\mathcal{W}_{2}^{(j)}\psi_{2s}=\psi_{2s}\>\>\forall\>\>j=1,2$. It can also be
verified in the same manner that
$\mathcal{W}_{2}^{(j)}\psi_{2s-1}\neq\pm\psi_{2s-1}$ and
$\mathcal{W}_{2}^{(j)}\psi_{2s}\neq\psi_{2s}$ $\>\>\forall\>\>j=1,2$ if the
eigenvalue has any non-zero imaginary part.

##  4 A special case of $B(c_{1},c_{2},i\alpha,i\alpha)$ related to
biorhogonal system and a deformed su(2) algebra

###### Definition 4

Two pairs of vectors $\\{|\phi_{j}\rangle:j=1,2\\}$ and
$\\{|\chi_{j}\rangle:j=1,2\\}$ are called bi-orthogonal if
$\langle\phi_{j}|\chi_{k}\rangle=0\>\>\forall\>\>j\neq k$[24, 39].

Considering a pair of orthogonal vectors
$\\{|u_{j}\rangle=\frac{1}{\sqrt{2}}\left(1\>\>(-1)^{j-1}\right)^{\dagger}:j=1,2\\}$,
$\langle u_{j}|u_{k}\rangle=\delta_{jk}$ we get the set of bi-orthogonal
vectors $|\phi_{j}\rangle=\omega T|u_{j}\rangle$ and
$|\chi_{j}\rangle=(T^{\dagger})^{-1}|u_{j}\rangle$ for $\omega=\cos\theta$ and
$j=1,2$. Here,
$T=\cos(\frac{\theta}{2})\mathbf{1}_{2}-2\sin(\frac{\theta}{2})\sigma_{2}$ and
$\omega=\sqrt{1-\alpha^{2}}$. This leads to a set of deformed $su(2)$
generators

$\displaystyle\sigma_{m}^{\alpha}=\frac{i^{m+1}}{2}\sum_{j,k=1}^{2}\frac{c_{jk}^{(m)}}{\omega^{\delta_{m2}}}|\phi_{j}\rangle\langle\chi_{k}|:m=1,2,3$
(11)

Here, $c_{jk}^{(1)}=(-1)^{j}\delta_{jk}$,
$c_{jk}^{(3)}=(-1)^{j}c_{jk}^{(2)}=(1-(-1)^{j}c_{jk}^{(1)})$.The generators
$\\{\sigma_{m}^{\alpha}:m=1,3\\}$ are non-hermitian in the conventional sense
of inner product in $\bf{C^{2}}$. The corresponding Jordan-Schwinger map is
given by

$\displaystyle
J_{m}^{\alpha}=\frac{i^{m-1}}{2}\sum_{j,k=1}^{2}[c_{jk}^{(4-m)}+i^{m}(1-\delta_{m2})\alpha
c_{jk}^{(m)}]a^{\dagger}_{j}a_{k}:m=1,2,3.$ (12)

The components of $J_{m}^{\alpha}$ have the following identifications :
$J_{1}^{\alpha}=B(-i\alpha,i\alpha,1,1)$, $J_{2}^{\alpha}=B(0,0,-i,i)$ and
$J_{3}^{\alpha}=B(1,-1,i\alpha,i\alpha)$. Introducing a new set of deformed
ladder operators with exponent $p$, $p\in\mathbf{R}$ (set of real numbers),
$J_{\pm}^{\alpha}=\sum_{r=0}^{1}(\pm i)^{r}\omega^{r-p}J^{\alpha}_{r+1}$ and
considering $J_{3}^{\alpha}=J_{0}^{\alpha}$ the following spectrum generating
algebra becomes possible

$[J_{0}^{\alpha},J_{\pm}^{\alpha}]=\pm\omega(\alpha)J_{\pm}^{\alpha}$ (13)
$[J_{+}^{\alpha},J_{-}^{\alpha}]=2\omega^{-2p+1}(\alpha)J_{0}^{\alpha}.$ (14)

It is to be noted that $J_{0}^{\alpha}$ is no longer hermitian and
$(J^{\alpha}_{+})^{\dagger}\neq J^{\alpha}_{-}$. The Killing metric tensor is
given by

$(g_{jk})=2\left(\begin{array}[]{ccc}\omega^{2}&0&0\\\ 0&0&2\omega^{-2p+2}\\\
0&2\omega^{-2p+2}&0\end{array}\right).$ (15)

As the Killing form is non-degenerate the algebra is semisimple. Casimir of
this algebra
${\mathcal{C}_{J}^{\alpha}}=\omega^{-2}J_{0}^{\alpha}(J_{0}^{\alpha}\pm\omega)+\omega^{2p-2}J^{\alpha}_{\mp}J_{\pm}^{\alpha}$.

###  4.1 Spectrum and eigenfunctions of $J_{0}^{\alpha}$

Writing $J_{0}^{\alpha}=\frac{1}{2}\mathcal{A}$ where, the entries of
$\mathcal{A}$ are equated with $\mathcal{M}$ with entries $\\{b_{0}\cdots
b_{l-1}\\}=\\{m,\cdots-m\\}$, $\\{c_{0}\cdots c_{l-2}\\}=\\{im\alpha,\cdots
i\alpha\\}$ and $\\{d_{0}\cdots d_{l-2}\\}=\\{i\alpha,\cdots im\alpha\\}$, the
spectrum of $\mathcal{A}$ is given by

$\displaystyle\Lambda^{\omega}_{odd}$ $\displaystyle=$
$\displaystyle\\{\lambda\\}_{odd}$ (16) $\displaystyle=$
$\displaystyle\\{-(2s-1)\omega,-(2s-3)\omega,\dots,(2s-3)\omega,(2s-1)\omega\\}$

for $m=2s-1.$ and

$\displaystyle\Lambda^{\omega}_{even}$ $\displaystyle=$
$\displaystyle\\{\lambda\\}_{even}=\\{-(2s)\omega,-(2s-2)\omega,\dots,(2s-2)\omega,(2s)\omega\\}$
(17)

for $m=2s.$

It is to be mentioned that the reality of the eigenvalues depends on the
parameter $\alpha$ and the condition of bi-orthogonality puts the restriction
$|\alpha|<1$ which fixes the eigenvalues on the real line. For $\alpha=1$ all
the eigenvalues coalesce to zero which becomes an exceptional point (for which
the geometric multiplicity of an eigenvalue is greater than its algebraic
multiplicity). If $\alpha$ is allowed to be treated as a free real parameter
there may occur a phase transition of eigenvalues from real to imaginary
values when $\alpha$ becomes greater than one. The eigenfunctions can be
obtained in view of section-3. It can be easily verified that
$\partial_{\mathcal{PT}}$ symmetry is completely lost for $\alpha>1$ as
expected. In the following we shall consider the cases with $m=2$ and $m=3$.

The eigenfunctions for $m=2$ corresponding to eigenvalues (of $\mathcal{A}$)
$-2\omega,0,2\omega$ can be given by

$\displaystyle\Psi_{-2}(\zeta_{1},\zeta_{2})$ $\displaystyle=$
$\displaystyle\zeta_{1}^{2}+2iG_{+}\zeta_{1}\zeta_{2}-G_{+}^{2}\zeta_{2}^{2}$
$\displaystyle\Psi_{0}(\zeta_{1},\zeta_{2})$ $\displaystyle=$
$\displaystyle\zeta_{1}^{2}+iG_{0}\zeta_{1}\zeta_{2}-\zeta_{2}^{2}$
$\displaystyle\Psi_{2}(\zeta_{1},\zeta_{2})$ $\displaystyle=$
$\displaystyle\zeta_{1}^{2}+2iG_{-}\zeta_{1}\zeta_{2}-G_{-}^{2}\zeta_{2}^{2}.$

Here, $G_{\pm}=\left(\frac{1\mp\omega}{1\pm\omega}\right)^{\frac{1}{2}}$ and
$G_{0}=\frac{2}{\sqrt{1-\omega^{2}}}$.

Similarly, for $m=3$ the eigenfunctions corresponding to the eigenvalues
$\\{-3\omega,-\omega,\omega,3\omega\\}$ are givan by

$\displaystyle\Phi_{-3}$ $\displaystyle=$
$\displaystyle\zeta_{1}^{3}+3iG_{+}\zeta_{1}^{2}\zeta_{2}-3G_{+}^{2}\zeta_{1}\zeta_{2}^{2}-G_{+}^{3}\zeta_{2}^{3}$
$\displaystyle\Phi_{-1}$ $\displaystyle=$
$\displaystyle\zeta_{1}^{3}+i(G_{0}+G_{+})\zeta_{1}^{2}\zeta_{2}+(G_{+}^{2}-G_{0}^{2}(1-\omega_{0}))\zeta_{1}\zeta_{2}^{2}-iG_{+}\zeta_{2}^{3}$
$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\zeta_{1}^{3}+i(G_{0}+G_{-})\zeta_{1}^{2}\zeta_{2}+(G_{-}^{2}-G_{0}^{2}(1+\omega_{0}))\zeta_{1}\zeta_{2}^{2}-iG_{-}\zeta_{2}^{3}$
$\displaystyle\Phi_{3}$ $\displaystyle=$
$\displaystyle\zeta_{1}^{3}+3iG_{-}\zeta_{1}^{2}\zeta_{2}-3G_{-}^{2}\zeta_{1}\zeta_{2}^{2}-G_{-}^{3}\zeta_{2}^{3}$

$\partial_{\mathcal{PT}}$ symmetry of the above eigenfunctions and possibility
of symmetry breaking can be understood in view of the discussion in section-3.
The wave functions are non-trivially modified in the sense that letting
$\gamma=0$ those corresponding to the hermitian case cannot be retrieved.

##  5 Conclusion

The above discussion makes it evident that (i) the construction of the
deformed algebra from bi-orthogonal vectors, (ii) reality of the eigenvalues
of $J_{0}^{\alpha}$ and (iii) existence and breaking of
$\partial_{\mathcal{PT}}$ symmetry of the eigenfunctions are interrelated.
Construction of a suitable symmetry induced inner product space may provide
further insights of such systems. Furthermore, the use of reproducing kernel
method is not much in vogue in conventional quantum methods. Investigation may
be furthered along this direction as well.

##  6 Acknowledgement

AC wishes to thank Dr. Baisakhi Mal for extending help in preparing the
document.

## References

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Delta-problems for the generalized Euler–Darboux equation

I. N. Rodionova, V. M. Dolgopolov, M. V. Dolgopolov

keywords: generalized Euler–Darboux equation, boundary value problem

Abstract. Degenerate hyperbolic equations are dealing with many important
issues for applied nature. While a variety of degenerate equations and
boundary conditions, successfully matched to these differential equation, most
in the characteristic coordinates reduced to Euler–Darboux one. Some boundary
value problems, in particular Cauchy problem, for the specified equation
demanded the introduction of special classes in which formulae are simple and
can be used to meet the new challenges, including Delta-problems in squares
that contain singularity line for equation coefficients with data on adjacent
or parallel sides of the square. In this short communication the generalized
Euler–Darboux equation with negative parameters in the rectangular region is
considered.

fa14072017 fb08092017 fc18092017

fe22092017

##  1 Introduction.
Statement of the problem and boundary conditions

Degenerate hyperbolic equations occur in many important problems of dynamical
systems and in questions of applied nature: the theory of infinitesimal
bending of surfaces of revolution, the membrane theory of shells, in plasma
magnetohydrodynamics, gas dynamics. With all the variety of degenerated
equations and boundary conditions it is successfully matched to the given
differential equation, the latter equation in the characteristic coordinates
reduces to the Euler–Darboux equations.

Some boundary value problems (Cauchy problem, in particular) for the specified
equations require the introduction of special classes in which the formula for
the solution becomes more simple in form and can be used to solve new tasks,
including Delta($\Delta$)-problems in the squares containing the singularity
line of the equation coefficients with the data on adjacent or parallel sides
of the square (directed by A. M. Nakhushev).

The first works on Delta($\Delta$)-problems on sets representing the union of
two characteristic triangles of hyperbolic equations were works of T. S.
Kalmenov [3], V. F. Volkodavov, A. A. Andreev [4], A. M. Nakhushev [5]. The
development of Delta-problems was done in the number of works of other
authors, from which we should mention [6, 7, 8, 9].

Unlike the previous in the present work the formulation of problems
$\Delta_{2}$ are on the set, which includes four of the characteristic
triangles. The generalized Euler–Darboux equation with negative parameters is
considered:

$U_{\xi\eta}-\frac{p}{\eta-(\mathop{\rm
sgn}\eta)\cdot\xi}U_{\xi}+\frac{p}{(\mathop{\rm
sgn}\eta)\cdot\eta-\xi}U_{\eta}-(\mathop{\rm sgn}\eta)\cdot\lambda U=0,$ (1)

Figure 1: *

Region $D$

$0<p<{1}/{2}$, $|\lambda|<\infty$ in the rectangular region $D$ bounded by
characteristics of equation (1) $\xi=0$, $\xi=h$, $\eta=h$, $\eta=-h$ ($h>0$),
containing within itself two lines of singularity of coefficients $\eta=\xi$
and $\eta=-\xi$. For equation (1) in the region $D$ the formulation of
boundary value problems $\Delta_{2}$ with the given values of the sought
solution on the parallel sides of the rectangle, with the conditions of
conjugation with respect to the solution and its normal derivative as lines of
singularity of the coefficients, and the internal characteristic of the line
are studied. The unique solvability of problems is proved by integral
equations method. Problems are solved in the special class of functions
introduced by authors in [10].

##  2 The solution of the problem

Considered boundary conditions for equation (1) in the rectangular region $D$
at the parallel sides of the rectangle are of the following form:

$U(\xi,h)=\varphi_{1}(\xi),\quad U(\xi,-h)=\varphi_{2}(\xi),\qquad
0\leq\xi\leq h.$

On lines of the singularity of $\eta=\xi$, $\eta=-\xi$ and on characteristic
$\eta=0$ the continuity of the sought solution is gluing on.

Relatively to normal derivatives two cases are considered of the pairing on
the lines $\xi=\pm\eta$. In the first case Frankl’s condition of occlusion is
imposed (Problem $\Delta_{2}^{*}$):

$\nu_{1}(\xi)=\lim_{\eta\to\xi+0}(\eta-\xi)^{-2p}(U_{\xi}-U_{\eta})=-\lim_{\eta\to\xi-0}(\xi-\eta)^{-2p}(U_{\xi}-U_{\eta})=-\nu_{2}(\xi),$
$\nu_{3}(\xi)=\lim_{-\eta\to\xi-0}(\eta+\xi)^{-2p}(U_{\xi}+U_{\eta})=-\lim_{-\eta\to\xi+0}(-\xi-\eta)^{-2p}(U_{\xi}+U_{\eta})=-\nu_{4}(\xi),$
$\leavevmode\nobreak\ \leavevmode\nobreak\ 0<\xi<h.$

In the second case the gluing is carried out on the continuity of normal
derivatives $\nu_{1}=\nu_{2}$; $\nu_{3}=\nu_{4}$ (Problem $\Delta_{2}$). In
both tasks, $\Delta_{2}$ and $\Delta_{2}^{*}$, in characteristic $\eta=0$
pairing sets:

$\lim_{\eta\to 0+0}\Bigl{(}\frac{\partial U}{\partial\eta}-\frac{\partial
U}{\partial\xi}\Bigr{)}=\lim_{\eta\to 0-0}\Bigl{(}\frac{\partial
U}{\partial\eta}+\frac{\partial U}{\partial\xi}\Bigr{)}.$

The basis for the decision of tasks in view is taken, obtained by the authors
[10, 11, 12], the solution of Cauchy problem of the special class $R_{h}$,
which is in one of four characteristic triangles that make up the region $D$,
has the form ($0<\xi<\eta<h$):

$U(\xi,\eta)=\int_{\eta}^{h}T_{1}(s)(s-\xi)^{p}(s-\eta)^{p}\,\,F_{0\hskip
8.53581pt1}\bigl{(}1+p;\lambda(s-\xi)(s-\eta)\bigr{)}ds+\\\
+\int^{\eta}_{\xi}N_{1}(s)(\eta-s)^{p}(s-\xi)^{p}\,\,F_{0\hskip
8.53581pt1}\bigl{(}1+p;-\lambda(\eta-s)(s-\xi)\bigr{)}ds,$ (2)

where $N_{1}=k_{1}T_{1}-k_{2}\nu_{1}$; $k_{1}$, $k_{2}=\rm const$;

$\,\,F_{0\hskip
8.53581pt1}\bigl{(}\alpha;z\bigr{)}=\sum_{n=0}^{\infty}\frac{z^{n}}{(\alpha)_{n}n!}.$

Formulas for Cauchy problem solutions in three other characteristic triangles
are not given. They also contain an unknown functions $T_{k}$, $N_{k}$,
$k=2,3,4$, which are searched in the class of continuous in the interval
$(0,h)$ and absolutely integrable functions on $[0,h]$.

The solution of the Problem $\Delta_{2}^{*}$ is reduced to the set of integral
equations of the form

$\int_{\xi}^{h}T(s)(\xi-s)^{p}\,\,F_{0\hskip 8.53581pt1}\bigl{(}1+p;\lambda
s(s-\xi)\bigr{)}ds=\Phi(\xi,\lambda),$

the unique solvability of which takes place when the following conditions are
imposed on the given functions:

$\varphi_{i}(\xi)\in
C^{(2)}[0,h],\qquad\varphi_{i}(0)=\varphi_{i}^{\prime}(0)=0,\qquad i=1,\,2,$
$\varphi_{i}(\xi)=(h-\xi)^{1+p+\varepsilon}\varphi_{i}^{\ast}(\xi),\qquad\varepsilon>0,\qquad\int_{0}^{h}\varphi_{i}(s)(h-s)^{-p-2}ds=0.$

When you run these conditions the only solution of the problem
$\Delta_{2}^{*}$ is given explicitly.

The complete study of the problem of $\Delta_{2}$ has managed to get only if
$\lambda=0$. In this case, its solution is reduced to a set of integral
equations of the first kind with Cauchy kernel:

$\int_{0}^{h}\frac{\mu_{i}(s)ds}{s-\xi}=\Phi_{i}^{\ast}(\xi),\quad
i=1,\,2;\qquad\mu_{1}=(h-s)^{p}s^{p}[\nu_{1}-\nu_{3}],$
$\mu_{2}=(h-s)^{p}s^{p-1}[\nu_{1}+\nu_{3}].$ (3)

${\Phi_{i}}^{\ast}(\xi)$ depends only on the given functions $\varphi_{i}$.

##  3 Discussions of solutions

Following the theory of singular integral equations [13], conditions imposed
on the given functions $\varphi$ under which there is the solution of
equations (3) (not unique), and solvability conditions of equations (3), and,
consequently, of the Problem $\Delta_{2}$.

  * 1.

If $\varphi_{i}(\xi)\in C[0,1]$, $\varphi_{i}^{\prime\prime}\in C(0,1)$,
$\varphi_{i}^{\prime\prime}$ are absolutely integrable on $[0,h]$, ${i=1,2}$,
then solutions of equations (3) have the form [13]:

$\mu_{1}=-\frac{1}{\pi}\sqrt{\frac{\xi}{h-\xi}}\int_{0}^{h}\sqrt{\frac{h-y}{y}}\cdot\frac{\Phi_{1}^{\ast}(y)}{y-\xi}dy,$
$\mu_{2}=-\frac{1}{\sqrt{\xi(h-\xi)}}\biggl{(}\frac{1}{\pi}\int_{0}^{h}\sqrt{(h-y)y}\cdot\frac{\Phi_{2}^{\ast}(y)}{y-\xi}dy+A_{0}\biggr{)},\quad
A_{0}=\rm const.$

  * 2.

If the existing conditions to add

$\varphi_{1}^{\prime}(h)+\varphi_{2}^{\prime}(h)=0,\quad\mbox{but}\leavevmode\nobreak\
\leavevmode\nobreak\ \varphi_{i}^{\prime}(h)\not=0,\quad i=1,\,2,$

then the second of equations (3) is uniquely solvable:

$\mu_{2}=-\frac{1}{\pi}\sqrt{\frac{h-\xi}{\xi}}\int_{0}^{h}\sqrt{\frac{y}{h-y}}\cdot\frac{\Phi_{2}^{\ast}(y)}{y-\xi}dy.$

In conclusion, we note that for the problem definition authors were inspired
by results, published in the work of I. V. Volovich, O. V. Groshev, N. A.
Gusev, E. A. Kuryanovich [14].

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