Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

Abstract

The matrix Sturm-Liouville operator on a finite interval with singular potential of class W −1 2 and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator.

Inverse spectral problems for functional-differential operators with involution

“…Inverse spectral problems for the matrix Sturm-Liouville equation (1.4) with W = I have been studied fairly completely (see [31][32][33][34][35][36][37][38]). Those results generalize the classical inverse problem theory for the scalar Sturm-Liouville equation −y ′′ + q(x)y = λy (see [23][24][25][26]).…”

“…Weyl functions and their generalizations are natural spectral characteristics in the inverse problem theory. In particular, the Weyl matrices have been used for reconstruction of the matrix Sturm-Liouville operators with W = I in [33][34][35][36][37][38].…”

“…In this paper, we do not rigorously prove the minimality of the given spectral data. However, for the matrix Sturm-Liouville operators with W = I, the complete spectral data characterization is obtained in [33][34][35]38]. Those spectral data are equivalent to the Weyl matrix.…”