Determination of the infinite Jacobi matrix with respect to two-spectra

G. Sh. Guseinov

Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 709-720.

Résumé

The inverse problem about two-spectra for the equation \begin{gather*} b_0y_0+a_0y_1=\lambda y_0,\\ a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}=\lambda y_n\qquad (n=1,2,3,\dots),\tag{1} \end{gather*} where $\{y_n\}_0^\infty$ is the desired solution, $\lambda$ is a complex parameter and $$ a_n>0, \quad\mathrm{Im}\,b_n=0,\qquad (n=0,1,2,\dots) $$ is studied. Necessary and sufficient conditions for the solvability of the inverse problem about two-spectra for Eq. (1) are established and also the procedure of reconstruction of the equation from its two-spectra is indicated.