Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function

G. Sh. Guseinov

Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 237-248

Résumé

Restoration from the generalized spectral function of the equations \begin{gather*} b_0y_0+a_0y_1=\lambda y_2, \\ a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}=\lambda y_n,\quad n=1,2,3,\dots, \end{gather*} where $a_n$ and $b_n$ are arbitrary complex numbers, $a_n\ne0$ ($n=0,1,2,\dots$), $\lambda$ is a complex parameter, and $\{y_n\}_0^\infty$ infin is the required solution, is investigated. Necessary and sufficient conditions for solvability of the inverse problem are obtained, and the restoration procedure is described.