Geodesic
On the spectral theory of dissipative difference operators of second order
Sbornik. Mathematics, Tome 66 (1990) no. 1, pp. 107-125

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The boundary conditions at infinity are used in a description of all maximal dissipative extensions of the minimal symmetric operator generated in the Hilbert space $l^2$ by the second-order difference expression $$ (\Lambda y)_n=a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1} $$ in the Weyl limit-circle case, where $n$ runs through the integer points on the half-line or the whole line, and the coefficients $a_n$ and $b_n$ are real. The characteristic functions of the dissipative extensions are computed. Completeness theorems are obtained for the system of eigenvectors and associated vectors. Bibliography: 13 titles. 
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B. P. Allakhverdiev; G. Sh. Guseinov. On the spectral theory of dissipative difference operators of second order. Sbornik. Mathematics, Tome 66 (1990) no. 1, pp. 107-125. http://geodesic.mathdoc.fr/item/SM_1990_66_1_a4/