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

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.