Understanding Partial $\mathcal{PT}$ symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space :An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

Non-hermitian Hamiltonian with real eihenvalues in the context of ${\mathcal{PT}}$ symmetry has become an interesting area of investigation for last couple of decades [1, 2, 3, 4, 5, 6, 7, 8]. The present article stems from a recent study of partial $\mathcal{PT}$ symmetry by Beygi et. al. [9] where a variable specific action of symmetry operator is understood considering a model of an N-coupled harmonic oscillator Hamiltonian with purely imaginary coupling terms. It has also been observed that the reality and partial reality of the spectrum have direct correspondences with the classical trajectories.The present formulation attempts to explore the possibility of partial $\mathcal{PT}$ symmetry in a Bose-Hubberd hamiltonian operator[10] as well as in its eigenstates in a typical Fock space environment. The relevant Fock space [11] has been viewed as a Reproducing Kernel Hilbert Space [12, 13, 14, 15, 16, 17]and the symetry operators are understood as Weighted Composition Conjugation [13, 14, 15] acting on it.

We begin with the following definition of Fock space involving functions of $n$ complex variable.

•       Definition 1.

  A Fock (or Segal-Bargmann) space $(\mathcal{F}^{2}({C}^{n}))$ is a separable complex Hilbert space of entire functions (of the complex variables $\{\zeta_{j}:j=1\dots n\}$ ) equipped with an inner-product

  	$\displaystyle\langle\psi,\phi\rangle=\prod_{j=1}^{n}\int_{W(\zeta_{j})}\psi(\zeta_{j}:j=1\dots n)\overline{\phi(\zeta_{j}:j=1\dots n)}$
  	$\displaystyle\;\;{\rm{with}}\prod_{j=1}^{n}\int_{W(\zeta_{j})}\equiv\int\int d{W(\zeta_{1})}\dots d{W(\zeta_{n})}.$		(1)

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

In a Fock space of one complex variable ${\mathcal{PT}}$ symmetry is often understood as a consequence of the more general notion of weighted composition conjugation [13] defined as follows.

•       Definition 2.

  Let, $\zeta$ 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 weighted composition conjugation is defined as

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

  (2)

The anti-linear operator $\mathcal{C}_{(\vartheta,\eta,\upsilon)}$ is a conjugation since it is involutive and isometric. The action of the operator $\mathcal{PT}$ is equivalent to the choice : $\vartheta=-1=-\upsilon,\eta=0$ which results to the following equation

$\displaystyle\mathcal{C}_{(\vartheta,0,1)}|_{\vartheta=-1}\psi(\zeta)=\overline{\psi(\overline{-\zeta})}.$

(3)

Similarly, the action of $\mathcal{T}$ is indicative of the choice : $\vartheta=1,\eta=0,\upsilon=1$ giving

$\displaystyle\mathcal{C}_{(\vartheta,0,1)}|_{\vartheta=1}\psi(\zeta)=\overline{\psi(\overline{\zeta})}.$

(4)

If $\psi$ is a function of several complex variables $\{\zeta_{j}:j=1\dots n\}$ one can define an operator $\mathcal{C}_{(\vartheta_{j},\eta_{j},\upsilon_{j}:j=1\dots n)}$ with the action

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

(5)

Let us introduce an operator $\mathcal{C}^{(j)}_{n}=\mathcal{C}_{(\vartheta_{j},\eta_{j}=0,\upsilon_{j}=1;j=1\dots n)}|_{\vartheta_{1}=1,\dots,\vartheta_{j}=-1,\dots,\vartheta_{n}=1}$ as $j$-th partial $\mathcal{PT}$ symmetry ($\partial_{\mathcal{PT}}$) operator through the following action

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

(6)

and a global $\mathcal{PT}$ symmetry operator $\mathcal{C}_{n}$ through the action

$\displaystyle\mathcal{C}_{n}\psi(\zeta_{1},\dots,\zeta_{j},\dots,\zeta_{n})=\overline{\psi(\overline{-\zeta_{1}},\dots,\overline{-\zeta_{j}},\dots,\overline{-\zeta_{n}})}.$

(7)

For our present purpose we shall only consider the operators $\mathcal{C}_{2}$ and $\{\mathcal{C}_{2}^{(j)}:j=1,2\}$. Now, global and partial $\mathcal{PT}$ symmetries of any function $\psi(\zeta_{1},\zeta_{2})$ are understood through the following equations

$\displaystyle\mathcal{C}_{2}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})\>\>{\rm and}\>\>\mathcal{C}_{2}^{(j)}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})\>\>\forall\>\>j=1,2$

(8)

respectively.

In the present discussion, following [10] a Bose-Hubbard type Hamiltonian has been considered. Such a Hamiltonian has been invoked as a two mode version for a second quantized many particle system showing Bose-Einstein Condensation (BEC) in a double well potential at low temperature. The said Hamiltonian becomes non-hermitian if one of the interaction terms present in it is taken as purely imaginary.The model Hamiltonian under consideration is given by

$\displaystyle H=\epsilon_{0}(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})+\epsilon(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1})+\alpha(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})^{2}.$

(9)

Here $\epsilon_{0}$ represents the on site energy difference, $\epsilon$ is the single particle tunneling and $\alpha$ stands for the interaction strength. $\{a_{j},a_{j}^{\dagger}:j=1,2\}$ are boson operators satisfying the condition $[a_{j},a^{\dagger}_{k}]-\delta_{jk}=[a_{j},a_{k}]=[a^{\dagger}_{j},a^{\dagger}_{k}]=0$. The Hamiltonian commutes with the number operator $N=a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}$ indicating particle conservation. For the time being we consider $\epsilon_{0}=1,\epsilon=i\gamma$ and $\gamma$ and $\alpha$ to be real.

In order to understand $\partial_{\mathcal{PT}}$ symmetry in Fock space we rewrite the Hamiltonian using Bargmann-Fock correspondence : $a_{j}^{\dagger}=\zeta_{j},a_{j}=\partial_{\zeta_{j}}$ as follows

$\displaystyle H=(\zeta_{1}\partial_{\zeta_{1}}-\zeta_{2}\partial_{\zeta_{2}})+i\gamma(\zeta_{1}\partial_{\zeta_{2}}+\zeta_{2}\partial_{\zeta_{1}})+\alpha(\zeta_{1}\partial_{\zeta_{1}}-\zeta_{2}\partial_{\zeta_{2}})^{2}.$

(10)

Following [13, 14, 15] we shall demonstrate the actions of weighted composition conjugations $\mathcal{C}_{2}$ and $\mathcal{C}_{2}^{(j)}$ on $H$ via the notion of Reproducing Kernel Hilbert Space (RKHS).

•       Definition 3.

  A function of the form $K^{[m_{j}]}_{\{\zeta_{j}\}}(u_{j}:j=1\dots n)=\prod_{j=1}^{n}u_{j}^{m_{j}}e^{u_{j}\overline{\zeta_{j}}}$($m_{j}\in{N}$ and $\zeta_{j},u_{j}\in{C}\>\>\forall\>\>j=1\dots n$) is called a kernel function (or a reproducing kernel) which satisfies the condition

  	$\displaystyle\psi^{(m_{1},m_{2},\dots,m_{n})}(\zeta_{1},\zeta_{2},\dots,\zeta_{n})=\langle\psi,K^{[m_{j}]}_{\{\zeta_{1}\}}\rangle$
  	$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}\dots\int_{W(u_{n})}\psi(u_{1},u_{2},\dots,u_{n})\overline{K^{[m_{j}]}_{\{\zeta_{1}\}}}.$		(11)

Considering the case with two complex variables the following proposition is immediate.

•       Proposition 1.

  $H^{\star}\neq H$ where $H^{\star}$ is defined as $\langle H\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle\psi(u_{1},u_{2}),H^{\star}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle$.

Proof : We shall first show that $\langle u_{1}\partial_{u_{1}}\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle\psi(u_{1},u_{2}),u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle$.

	$\displaystyle\langle u_{1}\partial_{u_{1}}\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle u_{1}\partial_{u_{1}}\psi{(u_{1},u_{2})},u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}\rangle$
	$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}u_{1}\partial_{u_{1}}\psi(u_{1},u_{2})\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}$
	$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}u_{1}[\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}}]\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}$
	$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})[\int_{W(u_{1})}\int_{W(u_{2})}u_{1}\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}}]$
	$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\langle u_{1}e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}},{u_{1}}^{m_{1}}{u_{2}}^{m_{2}}e^{{u_{1}}\overline{\zeta_{1}}+{u_{2}}\overline{\zeta_{2}}}\rangle$
	$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}(\zeta_{1}e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
	$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}\partial_{\overline{v_{1}}}(e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
	$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\psi(v_{1},v_{2})\overline{v_{1}}\partial_{\overline{v_{1}}}\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}(e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
	$\displaystyle=\langle\psi(v_{1},v_{2}),v_{1}\partial_{v_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(v_{1},v_{2})\rangle.$		(12)

An identical argument holds for $u_{2}\partial_{u_{2}}$ and similar calculations justify forms of adjoints for the operators $iu_{j}\partial_{u_{k}}$ for $j\neq k$. For example, it can be shown that

	$\displaystyle\langle iu_{1}\partial_{u_{2}}\psi(u_{1},u_{2}),u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}\rangle$
	$\displaystyle=\langle\psi(v_{1},v_{2}),-iv_{2}\partial_{v_{1}}v_{1}^{m_{1}}v_{2}^{m_{2}}e^{v_{1}\overline{\zeta_{1}}+v_{2}\overline{\zeta_{2}}}\rangle.$		(13)

Using these results in the expression of Hamiltonian the proposition is verified. 

•       Proposition 2.

  $H$ is $\mathcal{C}_{2}$ self-adjoint i. e.; $\mathcal{C}_{2}H^{\star}\mathcal{C}_{2}=H$

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

	$\displaystyle\mathcal{C}_{2}(u_{1}\partial_{u_{1}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
	$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}\mathcal{C}_{2}u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
	$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}\overline{\overline{(-u_{1})}^{m_{1}}}\overline{\overline{(-u_{2})}^{m_{2}}}\overline{e^{\overline{-u_{1}}\overline{\zeta_{1}}+\overline{-u_{2}}\overline{\zeta_{2}}}}$
	$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}(-1)^{m_{1}+m_{2}}u_{1}^{m_{1}}u_{2}^{m_{2}}e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}$
	$\displaystyle=\mathcal{C}_{2}u_{1}(-1)^{m_{1}+m_{2}}[u_{1}^{m_{1}}u_{2}^{m_{2}}(-{\zeta_{1}})e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}+m_{1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}]$
	$\displaystyle=-u_{1}(-1)^{m_{1}+m_{2}}[(-1)^{m_{1}+m_{2}}u_{1}^{m_{1}}u_{2}^{m_{2}}(-\overline{\zeta_{1}})$
	$\displaystyle+m_{1}(-1)^{m_{1}+m_{2}-1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}]e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
	$\displaystyle=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$		(14)

Similarly one can prove $\mathcal{C}_{2}(iu_{1}\partial_{u_{2}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=(iu_{2}\partial_{u_{1}})K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$ and $\mathcal{C}_{2}(iu_{2}\partial_{u_{1}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=(iu_{1}\partial_{u_{2}})K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$. Using these result in the expression of the Hamiltonian the above proposition is verified.  

•       Proposition 3.

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

Proof : We shall only verify the that $\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}\mathcal{C}_{2}^{(j)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$ through the following steps

	$\displaystyle\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}\mathcal{C}_{2}^{(1)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
	$\displaystyle=\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}[(-u_{1})^{m_{1}}(u_{2})^{m_{2}}e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}]$
	$\displaystyle=\mathcal{C}_{2}^{(1)}u_{1}(-1)^{m_{1}}[m_{1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}+(-\zeta_{1})u_{1}^{m_{1}}u_{2}^{m_{2}}]e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}$
	$\displaystyle=-u_{1}(-1)^{m_{1}}[m_{1}(-u_{1})^{m_{1}-1}u_{2}^{m_{2}}+\overline{(-\zeta_{1})}(-u_{1})^{m_{1}}u_{2}^{m_{2}}]e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
	$\displaystyle=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$		(15)

Similarly,

	$\displaystyle\mathcal{C}_{2}^{(1)}iu_{1}\partial_{u_{2}}\mathcal{C}_{2}^{(1)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
	$\displaystyle=\mathcal{C}_{2}^{(1)}iu_{1}\partial_{u_{2}}[(-u_{1})^{m_{1}}(u_{2})^{m_{2}}e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}]$
	$\displaystyle=\mathcal{C}_{2}^{(1)}iu_{1}(-1)^{m_{1}}[(u_{1})^{m_{1}}m_{2}(u_{2})^{m_{2}-1}+(u_{1})^{m_{1}}(u_{2})^{m_{2}}(\zeta_{2})]e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}$
	$\displaystyle=iu_{1}\partial_{u_{2}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$		(16)

Using these and similar results for the expression of the Hamiltonian the proposition can be verified. 

First we shall show that the reality of the eigenvalues is directly related to the $\partial_{\mathcal{PT}}$ symmetry of the eigen states of the Hamiltonian. It is readily observed that the present Hamiltonian leaves the homogeneous polynomial space of two indeterminates $(\zeta_{1},\zeta_{2})$ invariant. Considering such a space of degree of homogeneity $m$ and polynomial bases $\{f_{k}=\zeta_{1}^{m-k}\zeta_{2}^{k}:k=0\dots m\}$ the operator $H$ has the following tridiagonal representation

$\displaystyle H=\left(\begin{array}[]{cccccccc}\beta_{m}&i\gamma&0&0&0&\dots&0&0\\ im\gamma&\beta_{m-2}&2i\gamma&0&0&\dots&0&0\\ 0&i\gamma(m-1)&\beta_{m-4}&3i\gamma&0&\dots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\dots&\beta_{-(m-2)}&im\gamma\\ 0&0&0&0&0&\dots&i\gamma&\beta_{-m}\end{array}\right).$

(22)

Here, $\beta_{\mu}=\mu+\alpha{\mu}^{2}$. The the eigenvalues of such a matrix can be found out with the help of the following theorem [18].

•       Theorem 1.

  Given a tri-diagonal matrix

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

  (23)

  with $d_{i}\neq 0\>\>\forall\>\>i$, let us consider a polynomial $P_{n}(\lambda)$ that follows the well-known three term recursion relation [19, 20]

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

  (24)

  If $P_{-1}(\lambda)=0$ and $P_{0}(\lambda)=1$ the eigenvalues are given by the zeros of the polynomial $P_{l}(\lambda)$ and eigenvector corresponding to $j$-th eigenvalue $\lambda_{j}$ is given by the vector

  $$\left(\begin{array}[]{c}P_{0}(\lambda_{j})\\ P_{1}(\lambda_{j})\\ \vdots\\ P_{l-2}(\lambda_{j})\\ P_{l-1}(\lambda_{j})\end{array}\right).$$

  (25)

Proof : Let $\left(\begin{array}[]{c}v_{0}\\ v_{1}\\ \vdots\\ v_{l-2}\\ v_{l-1}\end{array}\right).$be the eigen vector corresponding to the eigenvalue $\lambda$. Then the eigenvalue equation gives us

	$\displaystyle b_{0}v_{0}+d_{0}v_{1}=\lambda v_{0}$
	$\displaystyle c_{0}v_{0}+b_{1}v_{1}+d_{1}v_{2}=\lambda v_{1}$
	$\displaystyle\vdots$
	$\displaystyle c_{l-2}v_{l-2}+b_{l-1}v_{l-1}=\lambda v_{l-1}$		(26)

Since $P_{0}(\lambda)=1$, one can write $v_{0}=P_{0}(\lambda)v_{0}$. Now in view of the recurrence relation (equation-24)

$\displaystyle v_{1}=\frac{1}{d_{0}}(\lambda-b_{0})P_{0}(\lambda)v_{0}=P_{1}(\lambda)$

(27)

Continuing the substitution recursively we get

$$v_{n}=P_{n}(\lambda)v_{0}:0\leq n<l$$

(28)

Substituting $n=l-1$ in equation-28 and using the three term relation (equation-24) we get

$$P_{l}(\lambda)v_{0}=0$$

(29)

giving the characteristic equation $P_{l}(\lambda)=0$. 

Now going back to the matrix in equation-22 and comparing the matrices in equation-22 and equation-23 we get $\{b_{0},\dots,b_{l-1}\}=\{\beta_{m},\dots,\beta_{-m}\}$, $\{c_{0},\dots,c_{l-2}\}=\{im\gamma,\dots,i\gamma\}$ and $\{d_{0},\dots,d_{l-2}\}=\{i\gamma,\dots,im\gamma\}$, one can determine the eigenvectors and eigenvalues of the matrix using the above algorithm.

The present investigation deals with a new kind of symmetry operator that acts on operators or functions in a variable specific way. In this article, the presence of such symmetry known as Partial $\mathcal{PT}$ symmetry is understood in a typical Fock space setting for a two-boson Bose-Hubbard type Hamiltonian with one purely imaginary interaction term. The symmetry operators are represented as Weighted Composition Conjugations acting on a Reproducing Kernel Hilbert Space. The existence of such symmetry as well as the possibility of breaking of such symmetry are found to have direct correspondence with the reality of eigenvalues of the Hamiltonian.