New Spectral Properties of Imaginary part of Gribov-Intissar Operator

TL;DR

This paper investigates new spectral properties of the imaginary part of the Gribov-Intissar operator, focusing on the right inverse on the negative imaginary axis and the deficiency indices of generalized Heun's operators, revealing conditions for their indeterminacy.

Contribution

It introduces novel spectral analysis results for the imaginary part of the Gribov-Intissar operator and characterizes the indeterminacy conditions of generalized Heun's operators.

Findings

Spectral properties of the right inverse on the negative imaginary axis are established.

Conditions for the generalized Heun's operator to be completely indeterminate are identified.

The generalized Heun's operator is shown to be entire of minimal type under certain parameters.

Abstract

In 1998, we have given in ([14] Intissar, A., Analyse de Scattering d'un op\'erateur cubique de Heun dans l'espace de Bargmann, Comm.Math.Phys.199 (1998) 243-256) the boundary conditions at infinity for a description of all maximal dissipative extensions in Bargmann space of the minimal Heun's operator $H_I = z(\frac{d}{dz} + z)\frac{d}{dz}$; $z \in \mathbb{C}$. The characteristic functions of the dissipative extensions have computed and some completeness theorems have obtained for the system of generalized eigenvectors. In ([18] Intissar, A, Le Bellac, M. and Zerner, M., Properties of the Hamiltonian of Reggeon field theory, Phys. Lett. B 113 (1982) 487-489) the non self-adjoint operator $\lambda H_{I}$ where $\lambda \in \mathbb{R}$ is imaginary part of the Hamiltonian of Reggeon field theory: $$H_{\mu, \lambda} = \mu z\frac{d}{dz} + i \lambda z( \frac{d}{dz} + z)\frac{d}{dz} \,\, \text{where} \,\, (\mu, \lambda) \in \mathbb{R}^{2} \,\, \text{and} \,\, i^{2} = -1$$ The main purpose of the present work is to present some new spectral properties of right inverse $K_{0, \lambda}$ of $H_{\lambda} = i\lambda H_I$ ($H_{\lambda}K_{0, \lambda} = I$) on negative imaginary axis and to study the deficiency numbers of the generalized Heun's operator $H^{p,m} = z^{p}( \frac{d^{m}}{dz^{m}} + z^{m})\frac{d^{p}}{dz^{p}}$ $p, m = 1, 2, ....$. In particular, here we find some conditions on the parameters $p$ and $m$ for that $H_{I}^{p,m}$ to be completely indeterminate.It follows from these conditions that $H^{p,m}$ is entire of the type minimal.