In the studies of acoustic waveguides in ocean, buckling of columns with variable cross sections in applied elasticity, transverse vibrations in nonhomogeneous strings, etc., we encounter a new class of problems of the type L1y1y1+q1(x)y1=λy1 defined on an interval [d1,d2] and L2y2y2+q2(x)y2=λy2 defined on the interval [d2,d3] satisfying certain matching conditions at the interface point x=d2.Earlier, in Part I, we constructed a fundamental system for (L1,L2) and derived certain estimates for the same.Here, in Part II, we consider four types of boundary value problems associated with (L1,L2) and study the corresponding spectra.