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.