Characterization of the spectrum of irregular boundary value problem for the Sturm-Liouville operator

TL;DR

This paper characterizes the spectrum of Sturm-Liouville operators with complex potentials and irregular boundary conditions, providing necessary and sufficient conditions for a set of complex numbers to be the spectrum.

Contribution

It introduces a complete spectral characterization for Sturm-Liouville problems with irregular boundary conditions and complex potentials, extending classical results.

Findings

Derived necessary and sufficient spectral conditions

Extended spectral theory to irregular boundary conditions

Provided criteria for spectrum realization

Abstract

We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.