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

In 1998, we gave a complete scattering analysis of the cubic Heun operator 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 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 Hp,m=a∗p(am+a∗m)ap;(p,m=1,2,…) acting on Bargmann space. In particular, here we find some conditions on the parameters p and m such that Hp,m is completely indeterminate. It follows from these conditions that Hp,m is entirely of minimal type. Then, we show that Hp,m and Hp,m+H∗p,m (where H∗p,m is the adjoint of Hp,m) are connected to the chaotic operators.