New Spectral Properties of Imaginary part of Gribov-Intissar Operator

In 1998, we have given in ([14] Intissar, A., Analyse de Scattering d’un opérateur 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 $H^{p,m}=z^p(\frac{d^m}{d{z}^m}+z^m)\frac{d^p}{d{z}^p}$ $p,m=1,2,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}$. 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.