Minimality conditions for Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Abstract

In the paper we study Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter. The necessary and sufficient conditions for minimality and completeness of the chosen system of root functions of the corresponding operator were given in two forms, one with the use of special associated functions and another one with the direct use of characteristic functions. This direct method was known for the affine case and was extensively discussed in the literature. The aim of the present paper is to develop this direct method for the quadratic case and to consider the affine and quadratic cases together in a unified way.