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In Communications in Mathematical Physics, no. 199, (1998), we haveconsidered the Heun operator $\displaystyle{ H = a^* (a + a^*)a}$ acting onBargmann space where $a$ and $a^{*}$ are the standard Bose annihilation andcreation operators satisfying the commutation relation $[a, a^{*}] = I$.  We have used the boundary conditions at infinity to give a description of allmaximal dissipative extensions in Bargmann space of the minimal Heun's operator$H$. The characteristic functions of the dissipative extensions have becomputed and some completeness theorems have be obtained for the system ofgeneralized eigenvectors of this operator.  In this paper we study the deficiency numbers of the generalized Heun'soperator  $\displaystyle{ H^{p,m} = a^{*^{p}} (a^{m} + a^{*^{m}})a^{p}; (p, m=1, 2,)}$ acting on Bargmann space. In particular, here we find some conditionson the parameters $p$ and $m$ for that $\displaystyle{ H^{p,m}}$ to becompletely 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 chaoticoperators. We will give a description of all maximal dissipative extensions andall selfadjoint extensions of the minimal generalized Heun's operator $H^{p,m}$acting on Bargmann space in separate paper.

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