On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

Abstract

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well.