On the complete indeterminacy and the chaoticity of generalized operator of Heun in Bargmann space

TL;DR

This paper investigates the properties of a generalized Heun operator in Bargmann space, focusing on its indeterminacy, dissipative extensions, and chaotic behavior, extending previous work on the classical Heun operator.

Contribution

It introduces conditions for the complete indeterminacy of the generalized Heun operator and explores its chaotic and dissipative extensions in Bargmann space.

Findings

Conditions for complete indeterminacy of H^{p,m}

H^{p,m} is entire of minimal type under certain parameters

H^{p,m} and its symmetrized form are connected to chaotic operators

Abstract

In Communications in Mathematical Physics, no. 199, (1998), we have considered the Heun operator $\displaystyle{ H = a^* (a + a^*)a}$ acting on Bargmann space where $a$ and $a^{*}$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a, a^{*}] = I$. We have 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 have be computed and some completeness theorems have be obtained for the system of generalized eigenvectors of this operator. In this paper we study the deficiency numbers of the generalized Heun's operator $\displaystyle{ H^{p,m} = a^{*^{p}} (a^{m} + a^{*^{m}})a^{p}; (p, m=1, 2, .....)}$ acting on Bargmann space. In particular, here we find some conditions on the parameters $p$ and $m$ for that $\displaystyle{ H^{p,m}}$ to be completely indeterminate. It follows from these conditions that $\displaystyle{ H^{p,m}}$ is entire of the type minimal. And we show that $\displaystyle{H^{p,m}}$ and $\displaystyle{ H^{p,m}+ H^{*^{p,m}}}$ (where $H^{*^{p,m}}$ is the adjoint of the $H^{p,m}$) are connected at the chaotic operators. We will give a description of all maximal dissipative extensions and all selfadjoint extensions of the minimal generalized Heun's operator $H^{p,m}$ acting on Bargmann space in separate paper.