A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations I

Abstract

In the studies of acoustic waveguides in ocean, buckling of columns with variable cross sections in applied elasticity, transverse vibrations in non homogeneous strings, etc., we encounter a new class of problems of the type L1y1=−d2y1dx2+q1(x)y1=λy1 defined on an interval [d1,d2] and L2y2=−d2y2dx2+q2(x)y2=λy2 on the adjacent interval [d2,d3] satisfying certain matching conditions at the interface point x=d2.Here in Part I, we constructed a fundamental system for (L1,L2) and derive certain estimates for the same. Later, in Part II, we shall consider four types of boundary value problems associated with (L1,L2) and study the corresponding spectra.