On the Complete Indeterminacy and the Chaoticity of the Generalized Heun Operator in Bargmann Space

Abstract

In 1998, we gave a complete scattering analysis of the cubic Heun operator ${\displaystyle H=a^{\ast }(a+a^{\ast })a}$ acting on Bargmann space, where a and $a^{\ast }$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a,a^{\ast }]=I$. We used the boundary conditions at infinity to give a description of all maximal dissipative extensions in Bargmann space of the minimal Heun’s operator H. The characteristic functions of the dissipative extensions were computed, and some completeness theorems were obtained for the system of generalized eigenvectors of this operator. In this paper, we study the deficiency numbers of the generalized Heun’s operator ${\displaystyle H^{p,m}=a^{{\ast }^p}(a^m+a^{{\ast }^m})a^p;(p,m=1,2,\dots )}$ acting on Bargmann space. In particular, here we find some conditions on the parameters p and m such that ${\displaystyle H^{p,m}}$ is completely indeterminate. It follows from these conditions that ${\displaystyle H^{p,m}}$ is entirely of minimal type. Then, we show that ${\displaystyle H^{p,m}}$ and ${\displaystyle H^{p,m}+H^{{\ast }^{p,m}}}$ (where $H^{{\ast }^{p,m}}$ is the adjoint of $H^{p,m}$) are connected to the chaotic operators.

1. Introduction and Preliminary Results

Let $\mathbb{B}$ be the Bargmann space [1] defined as a subspace of the space $O\left(\mathbb{C}\right)$ of holomorphic functions on $\mathbb{C}$, given by

$${\displaystyle \mathbb{B}=\left\{\varphi \in O\left(\mathbb{C}\right);⟨\varphi ,\varphi ⟩<\infty \right\},}$$

with the paring

$${\displaystyle ⟨\varphi ,\psi ⟩=\frac{1}{\pi }\underset{\mathbb{C}}{{\displaystyle \int }}\varphi \left(z\right)\overline{\psi \left(z\right)}d\mu \left(z\right)\forall \varphi ,\psi \in O\left(\mathbb{C}\right)}$$

where ${\displaystyle d\mu \left(z\right)=e^{-{\left|z\right|}^2}d\lambda \left(z\right)}$ is the Gaussian density on the complex plane $\mathbb{C}:=\{z=x+iy;(x,y)\in {\mathbb{R}}^2\}$, ${\left|z\right|}^2=x^2+y^2$ and $d\lambda \left(z\right)=dxdy$ is the usual Lebesgue measure on $\mathbb{C}={\mathbb{R}}^2$.

This space with $\mid \mid \varphi \mid {\mid }^2=⟨\varphi ,\varphi ⟩$ is a Hilbert space and ${\displaystyle e_k\left(z\right)=\frac{z^k}{\sqrt{k!}};k=0,1,\dots }$ is a complete orthonormal basis of $\mathbb{B}$.

Remark 1.

(i) The Bargmann space has a long history in mathematics and mathematical physics and has been given a wide variety of appellations, including many combinations and permutations of the names Bargmann, Fischer, Fock and Segal. Some papers of the above names are cited in [2].

(ii) In ([2]), the authors consider the Fock space of entire functions f such that

$${\displaystyle {\mathbb{F}}^p={\left[\frac{p}{2\pi }\underset{\mathbb{C}}{\int }{|f\left(z\right)e^{-\frac{1}{2}{\left|z\right|}^2}|}^{p}dxdy\right]}^{1/p}}<\infty ;0<p\le \infty ,$$

and for $m\in \mathbb{N}$ the space ${\mathbb{F}}^{(p,m)}$ consists of entire functions f such that $f^{\left(m\right)}$, the m-th order derivative of f, is in ${\mathbb{F}}^p$ (note that for $p=2$ we have ${\mathbb{F}}^2=\mathbb{B}$).

They established the following theorem:

Theorem 1  

([2]). Suppose $0<p\le \infty$, m is a positive integer and f is an entire function on $\mathbb{C}$. Then, $f\in {\mathbb{F}}^{p,m}$ if and only if the function $z^{m}f\left(z\right)$ is in ${\mathbb{F}}^p$.

(iii) Let $m=0$. Since the annihilation operator is unbounded, it is natural to ask the following question: can we find a condition on the derivative of f such that f is in $F^p$?

For $p=2$, a necessary and sufficient condition on $f^{\prime }$ is given in a lemma of the author presented in [3] (Lemme 1, 1984, page 127) which says that

If $f\in {\mathbb{F}}^2$ then ${\displaystyle {\int }_{\mathcal{C}}\frac{e^{-{\left|z\right|}^2}}{1+{\left|z\right|}^2}{\left|f^{\prime }\left(z\right)\right|}^{2}dxdy}$ is convergent.

This original lemma is recalled in reference [4]. In this reference, we can see another lemma ([4], Lemma 3, page 265) which also gives another original property of the Bargmann space.

If $f\in \mathbb{B}$, then its restrictions to subset $x+i\mathbb{R}$ are of square integrable with respect to $e^{-y^2}$ for all $x\in \mathbb{R}.$

In this article, the standard Bose operators for annihilation and creation are defined by

$$\left\{\begin{array}{c}a\varphi \left(z\right)={\varphi }^{\prime }\left(z\right)\hfill \\ \mathrm{with}\mathrm{maximal}\mathrm{domain}:\hfill \\ D\left(a\right)=\{\varphi \in \mathbb{B}suchthata\varphi \in \mathbb{B}\}.\hfill \end{array}\right.$$

(1)

$$\left\{\begin{array}{c}{a}^{\ast }\varphi \left(z\right)=z\varphi \left(z\right)\hfill \\ \mathrm{with}\mathrm{maximal}\mathrm{domain}:\hfill \\ D\left(a^{\ast }\right)=\{\varphi \in \mathbb{B}suchthat{a}^{\ast }\varphi \in \mathbb{B}\}.\hfill \end{array}\right.$$

(2)

Accordingly, for the operator $H^{p,m}=a^{{\ast }^p}(a^m+a^{{\ast }^m})a^p;(p,m=1,2,\dots )$ we have

$$\left\{\begin{array}{c}{H}^{p,m}\varphi \left(z\right)=z^p{\varphi }^{(p+m)}\left(z\right)+z^{p+m}{\varphi }^{\left(p\right)}\left(z\right)\hfill \\ \mathrm{with}\mathrm{maximal}\mathrm{domain}:\hfill \\ D\left(H_{max}^{p,m}\right)=\{\varphi \in \mathbb{B};H^{p,m}\varphi \in \mathbb{B}\}.\hfill \end{array}\right.$$

It follows from (1) and (2) that the action of the operator $a^{{\ast }^r}{a}^s$ ($r\in \mathbb{N}$, $s\in \mathbb{N}$) on an element $\varphi \in \mathbb{B}$; ${\displaystyle \varphi \left(z\right)=\sum _{k=0}^{\infty }{a}_k{e}_k\left(z\right)}$ is given by

${\displaystyle a^{{\ast }^r}{a}^s\varphi \left(z\right)}$ = ${\displaystyle \sum _{k=0}^{\infty }k(k-1)\dots (k-s+1)a_k\frac{z^{k+r-s}}{\sqrt{k!}}}$

${\displaystyle =\sum _{k=0}^{\infty }(k+s-r)(k+s-r-1)\dots (k-r+1)a_{k+s-r}\frac{z^k}{\sqrt{(k+s-r)!}}.}$

If $s\ge r$, we have ${\displaystyle \sqrt{(k+s-r)!}=\sqrt{k}\sqrt{(k+1)}\dots \sqrt{(k+s-r)}.}$ This implies that

${\displaystyle a^{{\ast }^r}{a}^s\varphi \left(z\right)}$ = ${\displaystyle \sum _{k=0}^{\infty }\{\sqrt{(k+s-r)}.\sqrt{(k+s-r-1)}\dots \sqrt{(k+1)}\dots (k-r+1)a_{k+s-r}\}{e}_k\left(z\right)}$.

Let ${\displaystyle u_k=\sqrt{(k+s-r)}.\sqrt{(k+s-r-1)}\dots \sqrt{(k+1)}\dots (k-r+1)}$. Then, we provide the following obvious lemma that we will use in the rest of this paper.

Lemma 1.

(i) 

If $s\ge r$, then ${\displaystyle u_k\sim k^{\frac{s+r}{2}}}$ i.e., ${\displaystyle u_k=O\left(k^{\frac{s+r}{2}}\right)}$.

(ii) 

Also, if $s\le r$, then ${\displaystyle u_k\sim k^{\frac{s+r}{2}}}$.

(iii) 

For $r=p$ and $s=p+m$, then ${\displaystyle u_k\sim k^{\frac{2p+m}{2}}}$.

Proof. 

As ${\displaystyle u_k\sim k^{\frac{s-r}{2}}{k}^r}$, then ${\displaystyle u_k\sim k^{\frac{s+r}{2}}}$. In a similar manner, we obtain $u_k$ in (ii) when $s\le r$ and (iii) is an obvious particular case of (i). The proof of Lemma 1 is complete. □

Now, the action of the operator $H^{p,m}$ on an element $\varphi \in \mathbb{B}$ is given by

${\displaystyle H^{p,m}\varphi \left(z\right)=\sum _{k=0}^{\infty }k(k-1)\dots (k-p-m+1)a_k\frac{z^{k-m}}{\sqrt{k!}}+}$

${\displaystyle \sum _{k=0}^{\infty }k(k-1)\dots (k-p+1)a_k\frac{z^{k+m}}{\sqrt{k!}}.}$

As $H^{p,m}$ is a polynomial in $(a^{\ast },a)$ of degree $2p+m$, we give the next Lemma, to study in separate paper, a perturbation of $H^{p,m}$ by self-adjoint operators of the form $\left(a^{{\ast }^j}{a}^j\right)$ with $j>p+{\displaystyle \frac{m}{2}}$. In particular we will show the non chaoticity of this perturbation and we will give a complete spectral analysis of these operators.

Lemma 2.

For all $j>p+{\displaystyle \frac{m}{2}}$, the following statement holds:

${\displaystyle \forall ϵ>0\exists C_{ϵ}>0}$ such that

${\displaystyle \forall \varphi \in D\left(a^{2j}\right)}$, ${\displaystyle \mid ⟨H^{p,m}\varphi ,\varphi ⟩\mid \le ϵ\mid \mid a^j\varphi \mid {\mid }^2+C_{ϵ}\mid \mid \varphi \mid {\mid }^2.}$

Proof. 

Let ${\displaystyle \varphi \in D\left(a^{2j}\right)}$ and ${\displaystyle u_k=\sqrt{(k+m)}.\sqrt{(k+m-1)}\dots \sqrt{(k+1)}\dots (k-p+1)}$. Then,

${\displaystyle ⟨a^{{\ast }^p}{a}^{p+m}\varphi ,\varphi ⟩=\sum _{k=0}^{\infty }{u}_k{a}_{k+m}{\overline{a}}_k}$

${\displaystyle \le \sum _{k=0}^{\infty }{u}_k\mid a_{k+m}\mid \mid {\overline{a}}_k\mid }$

${\displaystyle \le \frac{1}{2}\sum _{k=0}^{\infty }{u}_k\mid a_{k+m}{\mid }^2+\frac{1}{2}\sum _{k=0}^{\infty }{u}_k\mid a_k{\mid }^2}$

${\displaystyle \le \frac{1}{2}\sum _{k=0}^{\infty }(u_{k-m}+u_k)\mid a_k{\mid }^2.}$

By virtue of (iii) of Lemma 1, we have ${\displaystyle u_k\sim k^{\frac{2p+m}{2}}}$; then, there exist $c_0>0$ and $c_1>0$ such that ${\displaystyle u_k\le c_0+c_1{k}^{\frac{2p+m}{2}}.}$

Now, for $j>p+{\displaystyle \frac{m}{2}}$ we apply the Young’s inequality to obtain

${\displaystyle \forall \delta >0,\exists c_{\delta }>0\mathrm{such}\mathrm{that}{k}^{\frac{2p+m}{2}}\le \delta k^j+c_{\delta }.}$

This implies that

${\displaystyle ⟨a^{{\ast }^p}{a}^{p+m}\varphi ,\varphi ⟩\le c_1\delta \sum _{k=0}^{\infty }{k}^j\mid a_k{\mid }^2+(c_{\delta }+c_0)\sum _{k=0}^{\infty }\mid a_k{\mid }^2}$

• and

${\displaystyle \forall ϵ>0,\exists C_{ϵ}>0}$ such that ${\displaystyle \forall \varphi \in D\left(a^{2j}\right)}$,

${\displaystyle \mid ⟨a^{{\ast }^p}{a}^{p+m}\varphi ,\varphi ⟩\mid \le ϵ\mid ⟨a^{{\ast }^j}{a}^j\varphi ,\varphi ⟩\mid +C_{ϵ}\mid \mid \varphi \mid {\mid }^2}$.

As ${\displaystyle ⟨a^{{\ast }^{p+m}}{a}^p\varphi ,\varphi ⟩=⟨\varphi ,a^{{\ast }^p}{a}^{p+m}\varphi ⟩}$, we obtain

${\displaystyle \forall ϵ>0,\exists C_{ϵ}>0}$ such that ${\displaystyle \forall \varphi \in D\left(a^{2j}\right)}$,

${\displaystyle \mid ⟨H^{p,m}\varphi ,\varphi ⟩\mid \le ϵ\mid \mid a^j\varphi \mid {\mid }^2+C_{ϵ}\mid \mid \varphi \mid {\mid }^2}$.

The proof of Lemma 2 is complete. □

Now, the differential operators $a^{\ast }$ and a act on the function $e_k$ according to the following formulae:

$${\displaystyle a^{\ast }{e}_k=\sqrt{k+1}{e}_{k+1},a{e}_k=\sqrt{k}{e}_{k-1};e_{-1}=0,k=0,1,\dots }$$

(3)

It follows from (3) that

$$\left\{\begin{array}{c}{\displaystyle H^{p,m}{e}_k=0,\mathrm{if}k<p}\hfill \\ \hfill \\ {\displaystyle H^{p,m}{e}_k=\frac{\sqrt{k!(k+m)!}}{(k-p)!}{e}_{k+m},\mathrm{if}p\le k<p+m}\hfill \end{array}\right.$$

(4)

and

$${\displaystyle H^{p,m}{e}_k=\frac{\sqrt{k!(k-m)!}}{(k-p-m)!}{e}_{k-m}+\frac{\sqrt{k!(k+m)!}}{(k-p)!}{e}_{k+m}\mathrm{if}k\ge p+m.}$$

Thus, from (4), if we denote ${\displaystyle {\mathbb{B}}_p=\{\varphi \in \mathbb{B};\varphi \left(0\right)={\varphi }^{\prime }\left(0\right)=\dots ={\varphi }^{(p-1)}\left(0\right)=0\}}$ then this space is generated by ${\displaystyle \{e_p,e_{p+1},\dots \}}$ and the matrix representation of the minimal operator $\mathbb{H}$ generated by the expression $H^{p,m}$ acting on the basis ${\displaystyle e_k;k=p,p+1,\dots }$ is given by the symmetric Jacobi matrix $\mathbb{H}$ which has only two nonzero diagonals. Namely, its numerical entries are the matrices ${\displaystyle H_{ij}}$ of order m defined by

$$\mathbb{J}=\left\{\begin{array}{c}{\displaystyle H_{i,i}=H_{i,j}=O;\mathrm{if}\mid i-j\mid >1(i,j=1,2,\dots )}\hfill \\ WhereOisthezerom\times mmatrix\hfill \\ \hfill \\ and\hfill \\ \hfill \\ {\displaystyle H_{i+1,i}=H_{i,i+1}where{H}_{i,i+1}isdiagonalm\times mmatrix}\hfill \\ \hfill \\ suchthatitsnumericalentriesare\hfill \\ \hfill \\ {\displaystyle {\beta }_k^i=\frac{\sqrt{k!(k+m)!}}{(k-p)!};(i-1)m+1\le k\le im.}\hfill \end{array}\right.$$

(5)

Let ${\displaystyle A_i}$ and ${\displaystyle B_i=B_i^{\ast }}$$(i=1,2,\dots )$ be $m\times m$ matrices whose entries are complex numbers. Then, matrix (5) is a particular case of the infinite matrix whose general form is

$$\mathfrak{J}=\left(\begin{array}{cccccc}{A}_1& B_1& O& .& .& .\\ B_1^{\ast }& A_2& B_2& \ddots & .& .\\ O& B_2^{\ast }& A_3& B_3& \ddots & .\\ .& \ddots & \ddots & \ddots & \ddots & \ddots & \\ .& .& \ddots & \ddots & \ddots & \ddots & \\ .& .& .& \ddots & \ddots & \ddots & \end{array}\right).$$

where O is the zero $m\times m$ matrix and the asterisk denotes the adjoint matrix.

Let $l_m^2\left(\mathbb{N}\right)$ be the Hilbert space of infinite sequences $\varphi =({\varphi }_1,{\varphi }_2,\dots {\varphi }_i,\dots )$ with the inner product ${\displaystyle <\varphi ,\psi >=\sum _{i=1}^{\infty }{\varphi }_i{\overline{\psi }}_i}$ where ${\displaystyle {\varphi }_i=({\varphi }_i^1,{\varphi }_i^2,\dots ,{\varphi }_i^m)\in {\mathbb{C}}^m}$ and ${\displaystyle {\varphi }_i{\overline{\psi }}_i=\sum _{j=1}^m{\varphi }_i^j{\overline{\psi }}_i^j.}$

The matrix $\mathfrak{J}$ defines a symmetric operator $\mathfrak{T}$ in $l_m^2\left(\mathbb{N}\right)$ according to the formula

$${\displaystyle {\left(\mathfrak{T}\varphi \right)}_i=B_{i-1}{\varphi }_{i-1}+A_i{\varphi }_i+B_i{\varphi }_{i+1},i=1,2,\dots }$$

(6)

where ${\displaystyle {\varphi }_0=({\varphi }_0^1,{\varphi }_0^2,\dots ,{\varphi }_0^m)=(0,0,\dots ,0).}$

Then, for our operator, we have

$${\displaystyle {\left(H\varphi \right)}_i=H_{i,i-1}{\varphi }_{i-1}+H_{i,i+1}{\varphi }_{i+1},i=1,2,\dots }$$

(7)

where ${\displaystyle {\varphi }_0=({\varphi }_0^1,{\varphi }_0^2,\dots ,{\varphi }_0^m)=(0,0,\dots ,0).}$

Remark 2.

The closure $\mathbb{T}$ with domain $D\left(\mathbb{T}\right)$ of the operator $\mathfrak{T}$ is the minimal closed symmetric operator generated by expression (6) and the boundary condition ${\varphi }_0=0.$

Definition 1

(deficiency indices). Let $\mathcal{H}$ be a separable Hilbert space and denote by $<.>$ the inner product in this space. Let T be a closed symmetric operator densely defined in $\mathcal{H}$, i.e., $T\subset T^{\ast }$, with domain $D\left(T\right)\subset \mathcal{H}$.

(1) The deficiency indices $n_{+}\left(T\right)\mathrm{and}{n}_{-}\left(T\right)$ are defined as follows:

$n_{+}\left(T\right)$ is the dimension of ${\displaystyle M_z=(T-zI)D\left(T\right);\Im .z\ne 0}$; and $n_{-}\left(T\right)$ is the dimension of eigen-pace $N_z$ of T corresponding to the eigenvalue z of the operator T, i.e.,

$${\displaystyle n_{\pm }\left(T\right):=dim(\mathcal{H}\ominus range(T-zI))=dimker(T^{\ast }-zI),z\in {\mathbb{C}}_{\pm }}$$

(2) A closed operator T defined in $\mathcal{H}$ is said to be completely non-self-adjoint if there is no subspace reducing $\mathfrak{B}$ of $\mathcal{H}$ such that the part of T in this subspace is self-adjoint.

A completely non-self-adjoint symmetric operator is often referred to as simple.

Remark 3.

●₁

The theory of deficiency indices of closed symmetric operator in a complex Hilbert space is well known and well studied in [5].

●₂

Deficiency indices measure how far a symmetric operator is from being self-adjoint. Determining whether or not a symmetric operator is self-adjoint is important in physical applications because different self-adjoint extensions of the same operator yield different descriptions of the same system under consideration [5].

●₃

The deficiency indices $n_{+}\left(T\right)$ and $n_{-}\left(T\right)$ of a closed symmetric operator T in a complex Hilbert space $\mathcal{H}$ are also defined by

$${\displaystyle n_{\pm }\left(T\right):=dimker(T^{\ast }\pm iI);i^2=-1}$$

Definition 2

(completely indeterminate. Ref. [6]). A closed symmetric operator T acting on complex Hilbert space $\mathcal{H}$ is said completely indeterminate if $n_{+}\left(T\right)=n_{-}\left(T\right)\ne 0$.

Example 1

([7]). $n_{+}\left(H^{1,1}\right)=n_{-}\left(H^{1,1}\right)=1$.

Remark 4.

(i) According to Berezanskii, (see [8], Chapter VII, Section 2), it is well known that the deficiency numbers $n_{+}$ and $n_{-}$ of the operator $\mathbb{T}$ satisfy the inequalities $0\le n_{+}\le m$ and $0\le n_{-}\le m,$ where $n_{+}$ is the dimension of ${\mathfrak{M}}_z=(\mathbb{T}-zI)D\left(\mathbb{T}\right);\mathfrak{I}mz\ne 0$; and $n_{-}$ is the dimension of the eigensubspace ${\mathfrak{N}}_{\overline{z}}$ corresponding to the eigenvalue $\overline{z}$ of the operator $\mathbb{T}.$

(ii) According to Krein [6], the operator $\mathbb{T}$ is said to be completely indeterminate if $n_{+}=n_{-}=m$ and, for Kostyuchenko-Mirsoev [9], the completely indeterminate case holds for the operator $\mathbb{T}$ if and only if all solutions of the vector equation

$${\left(\mathfrak{T}\varphi \right)}_i=z{\varphi }_{i}i=1,2,\dots$$

belong to $l_m^2\left(\mathbb{N}\right)$ for $z=0.$

In 1949, Krein developed the theory of entire operators with arbitrary finite defect numbers; we refer to [6,9] and the references therein, which are closely connected with this theory. In Section 2, we give some properties associated with this theory for the generalized Heun’s operator ${\displaystyle H^{p,m}=a^{{\ast }^p}(a^m+a^{{\ast }^m})a^p}$, in particular its complete indeterminacy in Bargmann space.

In Section 3, we show that the operators ${\displaystyle \breve{\mathbb{H}}=a^{{\ast }^p}{a}^{p+m}\mathbb{U}}$, ${\displaystyle \breve{\mathbb{H}}+{\breve{\mathbb{H}}}^{\ast }}$, and $p=1,2,\dots ,m=1,2,\dots$ are chaotic where ${\displaystyle \mathbb{U}{e}_k=e_{k+m-1}}$ and ${\displaystyle {\breve{\mathbb{H}}}^{{\ast }^{p,m}}}$, ${\displaystyle {\mathbb{U}}^{\ast }}$ are the adjoint of $\breve{\mathbb{H}}$ and of ${\displaystyle \mathbb{U}}$, respectively.

2. On the Complete Indeterminacy of the Generalized Heun Operator in Bargmann Space

In [9], Kostyuchenko and Mirzoev gave some tests for the complete indeterminacy of a Jacobi matrix $\mathfrak{J}$ in terms of its entries $A_i$ and $B_i$. In the following, we give two lemmas which allow us to show the complete indeterminacy of the generalized Heun operator in Bargmann space.

For $m=1,2,\dots$, let ${\mathbb{C}}^m$ be the euclidean m-dimensional space and ${\displaystyle B_i=H_{i,i+1}}$ be the diagonal $m\times m$ matrix such that its numerical entries are

$${\displaystyle {\beta }_k^i=\frac{\sqrt{k!(k+m)!}}{(k-p)!};(i-1)m+1\le k\le im,i=1,2,\dots }$$

If we denote by $\mid \mid .\mid \mid$ the spectral matrix norm, then we obtain

$${\displaystyle \left(1\right)\mid \mid B_i\mid \mid =\frac{\sqrt{\left(im\right)!\left[\right(i+1\left)m\right]!}}{(im-p)!}\sim i^{p+\frac{m}{2}}}$$

$${\displaystyle \left(2\right)\mid \mid B_i^{-1}\mid \mid =\frac{1}{\mid \mid B_i\mid \mid }}$$

Lemma 3.

Let ${\displaystyle B_i=H_{i,i+1},(i=1,2,\dots )}$ be the diagonal $m\times m$ matrix ($m=1,2,\dots$) such that its numerical entries are

$${\displaystyle {\beta }_k^i=\frac{\sqrt{k!(k+m)!}}{(k-p)!};(i-1)m+1\le k\le im,i=1,2,\dots }$$

Then, the following inequality,

$${\displaystyle \mid \mid B_{i-1}\mid \mid .\mid \mid B_{i+1}\mid \mid \le \frac{1}{\mid \mid B_i^{-1}\mid {\mid }^2}}$$

(8)

holds starting from some $i\ge m.$

Proof. 

By using Lemma 1 or the behavior of gamma function $\mathsf{\Gamma }\left(x\right)$ as

$\mathcal{R}ex\to +\infty$ given by Stirling’s formula ${\displaystyle \mathsf{\Gamma }\left(x\right)\sim \sqrt{2\pi }{e}^{-x}{x}^{x-\frac{1}{2}}}$, we deduce that

${\displaystyle {\beta }_k^i\sim k^{p+\frac{m}{2}}}$ as $k\to +\infty$.

As ${\displaystyle \mid \mid B_i\mid \mid ={\beta }_{im}^i=\frac{\sqrt{\left(im\right)!\left(m\right(i+1\left)\right)!}}{(im-p)!}\sim {\left(im\right)}^{p+\frac{m}{2}}={\left(i\right)}^{p+\frac{m}{2}}{\left(m\right)}^{p+\frac{m}{2}}}$,

then ${\displaystyle \mid \mid B_{i-1}\mid \mid \sim {(i-1)}^{p+\frac{m}{2}}{\left(m\right)}^{p+\frac{m}{2}}}$, ${\displaystyle \mid \mid B_{i+1}\mid \mid \sim {(i+1)}^{p+\frac{m}{2}}{\left(m\right)}^{p+\frac{m}{2}}}$ and

${\displaystyle \mid \mid B_{i-1}\mid \mid }$.${\displaystyle \mid \mid B_{i+1}\mid \mid }$${\displaystyle \sim {\left(i\right)}^{2p+m}{(1-\frac{1}{i^2})}^{p+\frac{m}{2}}{\left(m\right)}^{2p+m}.}$

Now, as ${\displaystyle {(1-\frac{1}{i^2})}^{p+\frac{m}{2}}\le 1}$, then ${\displaystyle \mid \mid B_{i-1}\mid \mid }$${\displaystyle \mid \mid B_{i+1}\mid \mid \le }$${\displaystyle \mid \mid B_i\mid {\mid }^2}$ and as ${\displaystyle \mid \mid B_i^{-1}\mid \mid =\frac{1}{\mid \mid B_i\mid \mid }}$, then (8) holds. □

Lemma 4.

Let ${\displaystyle B_i=H_{i,i+1},(i=1,2,\dots )}$ be the diagonal $m\times m$ matrix, $(m=1,2,\dots )$ such that its numerical entries are

$${\displaystyle {\beta }_k^i=\frac{\sqrt{k!(k+m)!}}{(k-p)!};(i-1)m+1\le k\le im}$$

Then, if $2p+m>2$, the following inequality holds:

$${\displaystyle \sum _{i=1}^{+\infty }\frac{1}{\mid \mid B_i\mid \mid }<+\infty .}$$

(9)

Proof. 

As ${\displaystyle \mid \mid B_i\mid \mid \sim i^{p+\frac{m}{2}}}$, then if $p+{\displaystyle \frac{m}{2}}>1,$ the series ${\displaystyle \sum _{i=1}^{+\infty }\frac{1}{i^{p+\frac{m}{2}}}<+\infty }$. It follows that (9) holds. □

Now, we show the following theorem.

Theorem 2.

If $p+{\displaystyle \frac{m}{2}}>1$, then the operator $\mathbb{H}$ is completely indeterminate and its deficient numbers satisfy the conditions $n_{+}=n_{-}=m$.

Proof. 

By applying the results of Kostyuchenko and Mirsoev [9] to our operator, then the completely indeterminate case holds for the operator $\mathbb{H}$ if and only if all solutions of the vector equation

$$B_{i-1}{\varphi }_{i-1}+B_i{\varphi }_{i+1}=\lambda {\varphi }_i(i=1,2,\dots )\mathrm{belong}\mathrm{to}{l}_m^2\left(\mathbb{N}\right)\mathrm{for}\lambda =0,$$

where ${\displaystyle B_i=H_{i,i+1}}$ is the diagonal $m\times m$ matrix such that its numerical entries are given by ${\displaystyle {\beta }_k^i=\frac{\sqrt{k!(k+m)!}}{(k-p)!};(i-1)m+1\le k\le im.}$

Now, from (7), we consider the following system:

$${\displaystyle B_{i-1}{\varphi }_{i-1}+B_i{\varphi }_{i+1}=0(i=1,2,\dots )}$$

where ${\displaystyle {\varphi }_0=({\varphi }_0^1,{\varphi }_0^2,\dots ,{\varphi }_0^m)=(0,0,\dots ,0).}$

As ${\displaystyle B_i^{-1},(i=1,2,\dots )}$ exists we deduce that the solutions of the above equation have the following explicit form:

If $i=2j,(j=1,2,\dots )$ we have

${\displaystyle {\varphi }_{2j}=0}$ and ${\displaystyle {\varphi }_{2j+1}=-{\left(1\right)}^j{B}_{2j}^{-1}{B}_{2j-1}\times B_{2j-2}^{-1}{B}_{2j-3}\times \dots \times B_2^{-1}{B}_1{\varphi }_1.}$

This solution belongs to $l_2\left(\mathbb{N}\right)$ if ${\displaystyle \sum _{j=1}^{+\infty }\mid \mid B_{2j}^{-1}{B}_{2j-1}\times B_{2j-2}^{-1}{B}_{2j-3}\times \dots \times B_2^{-1}{B}_1\mid {\mid }^2<+\infty }$

and, if $i=2j-1,,(j=1,2,\dots )$, we have

${\displaystyle {\varphi }_{2j-1}=0}$ and ${\displaystyle {\varphi }_{2j}=-{\left(1\right)}^j{B}_{2j-1}^{-1}{B}_{2j-2}\times B_{2j-3}^{-1}{B}_{2j-4}\dots \times B_3^{-1}{B}_2{\varphi }_2.}$

This solution belongs to $l_2\left(\mathbb{N}\right)$ if ${\displaystyle \sum _{j=1}^{+\infty }\mid \mid B_{2j-1}^{-1}{B}_{2j-2}\times B_{2j-3}^{-1}{B}_{2j-4}\dots \times B_3^{-1}{B}_2\mid {\mid }^2<+\infty .}$

Then, the solution generated by the above equation belongs to $l_2\left(\mathbb{N}\right)$ if

$${\displaystyle \sum _{j=1}^{+\infty }\mid \mid B_{2j-1+ϵ}^{-1}{B}_{2j-2+ϵ}\times \dots \times B_{3+ϵ}^{-1}{B}_{2+ϵ}{B}_{1+ϵ}^{-1}{B}_{ϵ}\mid {\mid }^2<+\infty }$$

(10)

where $ϵ=0$ or $ϵ=1$ and ${\displaystyle B_0=B_1^{-1}.}$

Now, as

$${\displaystyle \mid \mid B_{2j-1+ϵ}^{-1}{B}_{2j-2+ϵ}\times \dots \times B_{3+ϵ}^{-1}{B}_{2+ϵ}{B}_{1+ϵ}^{-1}{B}_{ϵ}\mid {\mid }^2\le }$$

$${\displaystyle \mid \mid B_{2j-1+ϵ}^{-1}\mid {\mid }^2\mid \mid B_{2j-2+ϵ}\mid {\mid }^2\times \dots \times \mid \mid B_{3+ϵ}^{-1}\mid {\mid }^2\mid \mid B_{2+ϵ}\mid {\mid }^2\mid \mid B_{1+ϵ}^{-1}\mid {\mid }^2\mid \mid B_{ϵ}\mid {\mid }^2}$$

then it follows from (8) of Lemma 3 that

$$\begin{array}{l}\mid \mid B_{2j-1+ϵ}^{-1}\mid {\mid }^2\times \dots \times \mid \mid B_{3+ϵ}^{-1}\mid {\mid }^2\mid \mid B_{1+ϵ}^{-1}\mid {\mid }^2\le \\ {\displaystyle \frac{1}{\mid \mid B_{2j-2+ϵ}\mid {\mid }^2\times \dots \times \mid \mid B_{2+ϵ}\mid {\mid }^2\mid \mid B_{1+ϵ}^{-1}\mid {\mid }^2\mid \mid B_{ϵ}\mid \mid \mid \mid B_{2j+ϵ}\mid \mid }}\end{array}$$

and, consequently, the general term of series (10) does not exceed ${\displaystyle \frac{\mid \mid B_{ϵ}\mid \mid }{\mid \mid B_{2j+ϵ}\mid \mid }}$ and (9) of Lemma 4 ensures the convergence of the series ${\displaystyle \sum _{j=1}^{+\infty }\frac{1}{\mid \mid B_{2j+ϵ}\mid \mid }}$. The proof of the theorem is complete. □

Remark 5.

We conclude this section with a few observations.

(i) 

For a systematic study of the theory of entire operators, the following references can be consulted [10,11,12,13]:

(ii) 

For $p=m=1,H^{1,1}=a^{\ast }(a+a^{\ast })a$ is an imaginary part of the Hamiltonian of Reggeon field theory.

${\displaystyle \mathfrak{R}=\mu a^{\ast }a+i\lambda a^{\ast }(a+a^{\ast })a}$; $(\mu ,\lambda )\in {\mathbb{R}}^2$ and $i^2=-1$.

An asymptotic analysis of generalized eigenvectors of this operator is given in [14].

(iii) 

The cubic term $H^{1,1}$ parameterized by λ is analogous to the Lindbla operators for describing non-unitarity in open quantum systems.

(iv) 

It is well known that $H^{1,1}$ is symmetric but it is not self-adjoint, since its minimal domain does not coincide with its maximal domain.

(v) 

It is worth mentioning that a connection between the generalized Heun operator $H^{p,m}$ and Jacobi operators is used here for the first time.

(vi) 

The topic of the connection between concrete operators and the Jacobi operator is currently the subject of a great deal of research. See, for example, the recent references [15,16,17] on the topic of the connection of the Dirac operator and the Shrödinger operator.

Let ${\displaystyle \breve{\mathbb{H}}=a^{{\ast }^p}{a}^{p+m}\mathbb{U}}$$(p=1,2,\dots m=1,2,\dots ),$${\displaystyle {\breve{\mathbb{H}}}^{\ast }}$ be its adjoint and ${\displaystyle \mathbb{U}{e}_k=e_{k+m-1}.}$ In the next section, we study the chaoticity of the operators $\breve{\mathbb{H}}$ and ${\displaystyle \breve{\mathbb{H}}+{\breve{\mathbb{H}}}^{\ast }}$ on Bargmann space in the sense of following Devaney’s definition [18,19].

Definition 3.

A linear unbounded densely defined operator $(\mathbb{T},D(\mathbb{T}\left)\right)$ on a Banach space $\mathbb{X}$ is called chaotic if the following conditions are met:

(1) 

${\mathbb{T}}^n$ is closed for all positive integers, n;

(2) 

There exists an element ${\displaystyle \varphi \in D\left({\mathbb{T}}^{\infty }\right)=\bigcap _{n=1}^{\infty }D\left({\mathbb{T}}^n\right)}$ whose orbit,

$Orb(\mathbb{T},\varphi )=\{\varphi ,\mathbb{T}\varphi ,{\mathbb{T}}^2\varphi ,\dots \}$, is dense in $\mathbb{X}$, i.e., $\mathbb{T}$ is said to be hypercyclic;

(3) 

The set $\{\varphi \in \mathbb{X};\exists j\in \mathbb{N}$ such that ${\mathbb{T}}^j\varphi =\varphi \}$ of the periodic points of operator $\mathbb{T}$ are dense in $\mathbb{X}$.

Remark 6.

(i) 

It is well known that linear operators in finite-dimensional linear spaces cannot be chaotic but the nonlinear operator may be. Only in infinite-dimensional linear spaces can linear operators have chaotic properties.

These last properties are based on the phenomena of hypercyclicity or the phenomena of nonwandercity.

(ii) 

The study of the phenomena of hypercyclicity originates in the papers by Gulisashvili et al. [20], Birkhoff [21] and Maclane [22], which show, respectively, that the operators of translation and differentiation, acting on the space of entire functions, are hypercyclic.

(iii) 

Ansari asserts in [23] that powers of a hypercyclic bounded operator are also hypercyclic.

(iv) 

For an unbounded operator, Salas exhibited in [24] an unbounded hypercyclic operator whose square is not hypercyclic.

(v) 

Salas found in [25] an example of bilateral weighted shift $\mathbb{T}$ such that both $\mathbb{T}$ and ${\mathbb{T}}^{\ast }$ are hypercyclic. The operator $\mathbb{T}\oplus {\mathbb{T}}^{\ast }$ is not even cyclic and, therefore, the direct sum of hypercyclic operators is not always hypercyclic.

(vi) 

In Bargmann representation, the annihilation operator a is chaotic but $a+a^{\ast }$ is not chaotic where $a^{\ast }$ is its adjoint satisfying $[a,a^{\ast }]=I$.

(vii) 

The result of Salas shows that one must be careful in the formal manipulation of operators with restricted domains. For such operators it is often more convenient to work with vectors rather than with operators themselves.

3. On the Chaoticity of the Generalized Heun Operator in Bargmann Space

We begin by recalling some sufficient conditions on the hypercyclicity of unbounded operators given by the following B$\stackrel{`}{e}$s–Chan–Seubert Theorem.

Theorem 3

(B$\stackrel{`}{e}$s–Chan–Seubert [26], p. 258). Let $\mathbb{X}$ be a separable infinite-dimensional Banach space and let $\mathbb{T}$ be a densely defined linear operator on $\mathbb{X}$. Then, $\mathbb{T}$ is hypercyclic if

(i) 

${\mathbb{T}}^m$ is a closed operator for all positive integers, m.

(ii) 

There exist a dense subset $\mathbb{Y}$ of the domain $D\left(\mathbb{T}\right)$ of $\mathbb{T}$ and a (possibly nonlinear and discontinuous) mapping $\mathbb{S}:\mathbb{Y}⟶\mathbb{Y}$ so that $\mathbb{T}\mathbb{S}=I_{\mid \mathbb{Y}}$ ($I_{\mid \mathbb{Y}}$ is identity on $\mathbb{Y}$) and ${\mathbb{T}}^n,{\mathbb{S}}^n⟶0$ pointwise on $\mathbb{Y}$ as $n⟶\infty .$

Lemma 5.

Let $\mathbb{B}$ be the Bargmann space and a and $a^{\ast }$ be the annihilation and creation operators defined on $\mathbb{B}$ by $a\varphi \left(z\right)={\varphi }^{\prime }\left(z\right)$ and $a^{\ast }\varphi \left(z\right)=z\varphi \left(z\right)$. Then,

(1) 

$H^{0,m}=a^m+a^{{\ast }^m};m=1,2$ and $a^{{\ast }^p}{a}^p;p=1,2,\dots$ are not chaotic operators.

(2) 

a and $H^{1,1}=a^{\ast }(a+a^{\ast })a$ are chaotic operators.

Proof. 

(1)

In Bargmann representation, we note that

-

a and $a^{\ast }$ are non-self-adjoint operators but have same domain. The operators $a^m+a^{{\ast }^m};m=1,2,\dots$ are symmetric, then they are self-adjoint, and, consequently, they are not chaotic in Bargmann space.

-

The operators $a^{{\ast }^p}{a}^p;p=1,2,\dots$ are self-adjoint operators with compact resolvent; then, they are not chaotic in Bargmann space.

(2)

In [27], it is shown that $a^{{\ast }^p}{a}^{p+1}$ is chaotic for all $p\ge 0$; in particular, the operator a is chaotic on $\mathbb{B}$ and this result is generalized in [28] to $z^p{\mathbb{D}}^{p+1}$, where $\mathbb{D}$ is the Gelfond–Leontiev operator of a generalized differentiation [29] acting on generalized Fock–Bargmann space. In 2014, for some weighted shift $\mathbb{M}$ defined on $(\mathsf{\Gamma },\chi )$-$Theta$ Fock–Bargmann spaces, we showed in [30] that the operators ${\mathbb{M}}^p{\mathbb{D}}^{p+1}$ are chaotic for all $p\ge 0$ where $\mathbb{D}$ is adjoint of $\mathbb{M}$.

It is proved in [31] (Theorem 2.3, Section 2, page 494) that $H^{1,1}=a^{\ast }(a+a^{\ast })a$ is chaotic on $\mathbb{B}$. This last operator plays an essential role in Reggeon field theory (see [4]). Also, we can show that $H^{1,1}=a^{\ast }(a+a^{\ast })a$ is chaotic on $\mathbb{B}$ by choosing ${\gamma }_n=\sqrt{n}ln\left(n\right)$ in the following Theorem [31] on the chaoticity of the sum of chaotic shifts with their adjoint in Hilbert space.

Theorem 4

([31]). Let a linear unbounded densely defined chaotic shift operator $(\mathbb{T},D(\mathbb{T}\left)\right)$ on a Hilbert space $\mathbb{E}=${${\displaystyle {\displaystyle \varphi ;\varphi =\sum _{n=1}^{\infty }{a}_n{e}_n}}$}such that its adjoint is defined by

$${\mathbb{T}}^{\ast }{e}_n={\omega }_n{e}_{n+1}$$

where $\left\{e_n\right\}$ is an orthonormal basis of $\mathbb{E}$ and ${\omega }_n$ is a positive weight associated with $\mathbb{T}.$

We assume that

$${\displaystyle \left(AssumptionHy{p}_1\right)\sum _{n=1}^{\infty }\frac{1}{{\omega }_n}<\infty ,}$$

(11)

$${\displaystyle \left(AssumptionHy{p}_2\right){\omega }_{n-1}{\omega }_{n+1}\le {\omega }_n^2,}$$

(12)

In $AssumptionHy{p}_3,\mathrm{there}\mathrm{exist}\alpha >0,\beta >0,a>0$ and a sequence ${\gamma }_n$ such that

$${\displaystyle \left(1\right)\frac{{\omega }_n{\gamma }_n}{{\gamma }_{n+1}}\ge n^{1+\alpha },}$$

(13)

$${\displaystyle \left(2\right)\frac{{\omega }_{n-1}{\gamma }_{n+1}}{{\omega }_n{\gamma }_{n-1}}=1-\frac{a}{n}+O\left(\frac{1}{n^{1+\beta }}\right),}$$

(14)

and

$${\displaystyle \left(3\right)\sum _{k=1}^{\infty }\frac{1}{{\gamma }_n^2}<\infty .}$$

(15)

Then, we have the following:

(i) 

For $\lambda \in \mathbb{C}$ the following recurrence sequence

$$(\ast )\left\{\begin{array}{c}{\displaystyle u_1\left(\lambda \right)=1}\hfill \\ {\displaystyle u_2\left(\lambda \right)=\frac{\lambda }{{\omega }_1}}\hfill \\ {\displaystyle {\omega }_{n-1}{u}_{n-1}\left(\lambda \right)+{\omega }_n{u}_{n+1}\left(\lambda \right)=\lambda u_n\left(\lambda \right)}\hfill \end{array}\right.$$

is solvable for all $\lambda \in \mathbb{C}$.

(ii) 

${\displaystyle \sum _{n=1}^{\infty }\mid u_n\left(\lambda \right){\mid }^2<\infty }$ for all $\lambda \in \mathbb{C}$.

(iii) 

The spectrum of $\mathbb{T}+{\mathbb{T}}^{\ast }$ is the all complex plane $\mathbb{C}$.

(iv) 

${(\mathbb{T}+{\mathbb{T}}^{\ast })}^m$ is closed, $\forall m\in \mathbb{N}$.

(v) 

$\mathbb{T}+{\mathbb{T}}^{\ast }$ is a hypercyclic operator.

(vi) 

$\mathbb{T}+{\mathbb{T}}^{\ast }$ is a chaotic operator.

Remark 7.

(i) 

In Bargmann representation, operator a gives an example of a linear unbounded densely defined chaotic shift operator $(\mathbb{T},D(\mathbb{T}\left)\right)$ on a Hilbert space $\mathbb{B}$ such that $\mathbb{T}+{\mathbb{T}}^{\ast }$ is not a chaotic operator.

(ii) 

In Bargmann representation, let $\mathbb{T}$ be a linear unbounded densely defined chaotic shift operator such that $n_{+}=n_{-}\ne 0$, where $n_{+}$ and $n_{-}$ are the defect numbers of $\mathbb{T}+{\mathbb{T}}^{\ast }$. Then, is the $\mathbb{T}+{\mathbb{T}}^{\ast }$ operator chaotic?

Now, we define $\mathbb{U}$ by

$$\mathbb{U}{e}_k=e_{k+m-1}$$

and its adjoint ${\mathbb{U}}^{\ast }$ by

$${\mathbb{U}}^{\ast }{e}_k=e_{k-m+1};k\ge m.$$

Let ${\displaystyle \breve{\mathbb{H}}=a^{{\ast }^p}{a}^{p+m}\mathbb{U}}$ and ${\displaystyle {\breve{\mathbb{H}}}^{\ast }={\mathbb{U}}^{\ast }{a}^{{\ast }^{p+m}}{a}^p}$.

Then,

$${\displaystyle \breve{\mathbb{H}}{e}_k={\omega }_{k-1}^{p,m}{e}_{k-1}}$$

where

$${\displaystyle {\omega }_{k-1}^{p,m}=\frac{\sqrt{(k-1)!(k-1+m)!}}{(k-1-p)!}}$$

and

$${\displaystyle {\breve{\mathbb{H}}}^{\ast }{e}_k={\omega }_k^{p,m}{e}_{k+1}.}$$

In the following, we show that the operator ${\displaystyle \breve{\mathbb{H}}}$ is chaotic and we apply Theorem 3 to prove that ${\displaystyle \breve{\mathbb{H}}+{\breve{\mathbb{H}}}^{\ast }}$ is also chaotic.

Theorem 5.

Let ${\mathbb{B}}_p,p=0,1,\dots$ be the subspace of Bargmann space generated by ${\displaystyle e_k;.k\ge p}$ and the operator ${\displaystyle \breve{\mathbb{H}}}$ with domain ${\displaystyle D\left(\breve{\mathbb{H}}\right)=\{\varphi \in {\mathbb{B}}_p;\breve{\mathbb{H}}\varphi \in {\mathbb{B}}_p\}}$ be defined by

${\displaystyle \breve{\mathbb{H}}{e}_k={\omega }_{k-1}^{p,m}{e}_{k-1}}$ where ${\displaystyle {\omega }_{k-1}^{p,m}=\frac{\sqrt{(k-1)!(k-1+m)!}}{(k-p)!}}$

Then,

${\displaystyle \breve{\mathbb{H}}}$ is chaotic on ${\mathbb{B}}_p$.

Proof. 

To use the Theorem of B$\stackrel{`}{\mathrm{e}}$s et al. [26] we begin by observing that for ${\displaystyle \varphi \left(z\right)=\sum _{k=p}^{\infty }{a}_k{e}_k\left(z\right)}$ such that ${\displaystyle \sum _{k=p}^{\infty }\mid a_k{\mid }^2<\infty }$ we have the following obvious properties:

(i)

${\displaystyle {\breve{\mathbb{H}}}^l\varphi \left(z\right)=\sum _{k=p}^{\infty }[\prod _{j=p}^{l+k-1}{\omega }_j^{p,m}]a_{k+l}{e}_k\left(z\right)}$ of domain

${\displaystyle D\left({\breve{\mathbb{H}}}^l\right)=\{\varphi =\sum _{k=p}^{\infty }{a}_k{e}_k;\sum _{k=p}^{\infty }\mid a_k{\mid }^2<\infty and\sum _{k=p}^{\infty }{[\prod _{j=p}^{l+k-1}{\omega }_j^{p,m}]}^2\mid a_{k+l}{\mid }^2<+\infty \}}$

is dense in ${\displaystyle {\mathbb{B}}_p\forall l\in \mathbb{N}}$.

(ii)

${\breve{\mathbb{H}}}^l$ is closed $\forall l\in \mathbb{N}$ and ${\breve{\mathbb{H}}}^l{e}_k\left(z\right)=0.$$\forall l>k\ge p\ge 0$.

(iii)

As ${\omega }_n^{p,m}\to +\infty$, then the spectrum of $\breve{\mathbb{H}}$ is all the complex plane.

In fact, let ${\displaystyle {\varphi }_{\lambda }=\sum _{k=p}^{\infty }{a}_k{e}_k}$ with ${\displaystyle a_k=\prod _{j=p}^{k-1}\frac{\lambda }{{\omega }_j^{p,m}}}$, i.e.,

${\displaystyle {\varphi }_{\lambda }=\sum _{k=p}^{+\infty }[\prod _{j=p}^{k-1}\frac{\lambda }{{\omega }_j^{p,m}}]e_k}$; then, as $a_p=0$, we deduce that

$$\breve{\mathbb{H}}{\varphi }_{\lambda }=\lambda {\varphi }_{\lambda },\forall \lambda \in \mathbb{C}$$

and as ${\displaystyle \sum _{k=p}^{\infty }{[\prod _{j=p}^k\frac{\lambda }{{\omega }_j^{p,m}}]}^2<+\infty }$, then ${\varphi }_{\lambda }\in D\left(\breve{\mathbb{H}}\right).$

Now, take $\mathbb{Y}$ the linear subspace generated by finite combinations of basis ${\left\{e_k\right\}}_{k=p}^{\infty }$; this subspace $\mathbb{Y}$ is dense in ${\displaystyle {\mathbb{B}}_p}$ and we define on it the operator $\mathbb{S}$ acting on ${\displaystyle \varphi =\sum _{k=p}^N{a}_k{e}_k}$ as follows:

$${\displaystyle \mathbb{S}\varphi =\sum _{k=p}^{N+1}\frac{a_{k-1}}{{\omega }_{k-1}^{p,m}}{e}_k}$$

and

$${\displaystyle {\mathbb{S}}^n{e}_k=\frac{1}{{\displaystyle \prod _{j=k}^{n+k}{\omega }_j^{p,m}}}{e}_{k+n}}$$

As ${\displaystyle \prod _{j=p}^n{\omega }_j^{p,m}\to +\infty }$ as $n\to +\infty$, we deduce that

$${\displaystyle {\mathbb{S}}^n{e}_k\to 0\mathrm{in}{\mathbb{B}}_p\mathrm{as}n\to +\infty ,k\mathrm{any}.}$$

By noting that ${\breve{\mathbb{H}}}^n{e}_k=0$ for $n>k$ and any element of $\mathbb{Y}$ can be annihilated by a finite power of $\breve{\mathbb{H}}$ and $\breve{\mathbb{H}}\mathbb{S}={\mathbb{I}}_{\mid \mathbb{Y}}$, then the hypercyclic nature of $\breve{\mathbb{H}}$ follows from the above Theorem of $B\stackrel{`}{e}s$ et al. [26].

We shall now show that $\breve{\mathbb{H}}$ has a dense set of periodic points.

To see this, it suffices to show that for every element $\varphi$ in the dense subspace $\mathbb{Y}$ there is a periodic point $\psi$ arbitrarily close to it.

For $s\ge p$ and $N\ge s$, we put

$${\displaystyle {\phi }_{s,N}\left(z\right)=e_s\left(z\right)+\sum _{k=s+1}^{\infty }\left[\prod _{j=s}^{kN+s-1}\frac{1}{{\omega }_j^{p,m}}\right]e_{kN+s}\left(z\right).}$$

We can observe the following obvious properties. □

Lemma 6.

(i) 

${\displaystyle {\breve{\mathbb{H}}}^N\prod _{j=0}^{kN-1}\frac{1}{{\omega }_j^{p,m}}{e}_{kN}=\prod _{j=0}^{(k-1)N-1}\frac{1}{{\omega }_j^{p,m}}{e}_{(k-1)N}\forall k\ge p}$.

(ii) 

${\displaystyle {\breve{\mathbb{H}}}^N\prod _{j=s}^{kN-1+s}\frac{1}{{\omega }_j^{p,m}}{e}_{kN+s}=\prod _{j=s}^{(k-1)N-1}\frac{1}{{\omega }_j^{p,m}}{e}_{(k-1)N+s}}$ for $s\ge p$, $N\ge s$ and $k\ge p.$

(iii) 

${\displaystyle {\phi }_{s,N}}$ is an N-periodic point of $\breve{\mathbb{H}}$.

(iv) 

${\displaystyle {\phi }_{s,N}\in D\left({\breve{\mathbb{H}}}^N\right)}$.

Now, let

$${\displaystyle \varphi \left(z\right)=\sum _{s=p}^M{a}_s{e}_s\left(z\right),}$$

such that

$${\displaystyle \left|a_s\prod _{j=p}^{s-1}{\omega }_j^{p,m}\right|<1;s=p,p+1,\dots ,M,}$$

and we choose the periodic point for $\breve{\mathbb{H}}$ as $\psi \left(z\right)$

$${\displaystyle \psi \left(z\right)=\sum _{s=p}^M{a}_s{\phi }_{s,N}\left(z\right),}$$

Then, there exists $N\ge M$ such that

$$\mid \mid \varphi -\psi \mid \mid \le ϵ\forall ϵ>0.$$

Remark 8.

We also can use the results of Bermudez et al. [32] to prove the chaoticity of our operator ${\displaystyle \breve{\mathbb{H}}}$.

Theorem 6.

Let ${\mathbb{B}}_p$ be the subspace of Bargmann space generated by ${\displaystyle e_k;k\ge p}$. Then,

${\displaystyle \breve{\mathbb{H}}+{\breve{\mathbb{H}}}^{\ast }}$ is chaotic where ${\displaystyle {\breve{\mathbb{H}}}^{\ast }}$ is the adjoint operator of ${\displaystyle \breve{\mathbb{H}}.}$

Proof. 

For ${\displaystyle {\omega }_k^{p,m}=\sqrt{k!}\frac{\sqrt{(k+m)!}}{(k-m)!}\sim k^{p+\frac{m}{2}}}$, assumptions (11), (12) and (15) of Theorem 3 hold. The validity of assumptions (13) and (14) remains to be checked.

(i)

Assumption (13) is valid. In fact, as ${\displaystyle {\omega }_k^{p,m}=\sqrt{k!}\frac{\sqrt{(k+m)!}}{(k-m)!}\sim k^{p+\frac{m}{2}}}$ then ${\displaystyle \frac{{\omega }_{k-1}^{p,m}}{{\omega }_k^{p,m}}=}$${\left(1-{\displaystyle \frac{1}{k}}\right)}^{p+{\displaystyle \frac{m}{2}}}=1-{\displaystyle \frac{p+\frac{m}{2}}{k}}+O\left({\displaystyle \frac{1}{k^2}}\right)$.

(ii)

Assumption (14) is valid. In fact, if we choose ${\displaystyle {\gamma }_k=k^{\frac{m}{2}}ln\left(k^p\right)=p{k}^{\frac{m}{2}}ln\left(k\right)}$ then ${\displaystyle \frac{{\gamma }_k}{{\gamma }_{k+1}}=1-\frac{1+\frac{m}{2}}{k}+O\left(\frac{1}{k^2}\right)}$

and for $p+{\displaystyle \frac{m}{2}}>2$ we obtain ${\displaystyle {\omega }_k\frac{{\gamma }_k}{{\gamma }_{k+1}}\ge k^{1+\beta }}$ where $1+\beta =p+{\displaystyle \frac{m}{2}}-1$.

□

4. Conclusions

We conclude this paper by summarizing the results obtained. In Bargmann space, we studied the generalized Heun operator ${\displaystyle H^{p,m}=a^{{\ast }^p}(a^m+a^{{\ast }^m})a^p}$ such that $(p,m=1,2,\dots )$ and $[a,a^{\ast }]=I$. In the case where $p=m=1$, this operator characterizes the cubic interaction operator of the Reggeon field theory. We have found some conditions of the parameters p and m for $H^{p,m}$ to be completely indeterminate; in particular, it is entirely of a minimal type. By using the results of reference [31], we have shown that $H^{p,m}$ and $H^{p,m}+H^{{\ast }^{p,m}}$ (where $H^{{\ast }^{p,m}}$ is the adjoint of $H^{p,m}$) are connected to the chaotic operators.

Based on the work of this paper as well as that of reference [7], we will give a description of all maximal dissipative extensions and all self-adjoint extensions of the minimal generalized Heun operator $H^{p,m}$ acting on Bargmann space in a separate paper.

It is worth mentioning that a connection between the generalized Heun operator $H^{p,m}$ and Jacobi operators is used here for the first time. We can add this example to the list of examples of reference [33].