Coercive Solvability of Many-Interval Sturm-Liouville Problems

It is well-known that the Sturm-Liouville type boundary value problems appears in solving many important problems in physics. For example, some problems of theory of elasticity in a half-strip ,the problems of theory of vibrations of an elastic cylinder reduce to investigation of solvability of appropriate boundary value problems for Sturm-Liouville type differential equations. In the recent years, the Sturm-Liouville type boundary value problems with additional transmission conditions are investigated by many mathematical and physical researches . Note that, such type problems is very complicated because boundary value problems with additional transmission conditions may be not self-adjoint in the classical Hilbert space L2 and therefore the eigenvalues may be not real. The main goal of this study is to prove coerciveness of a new class many interval Sturm-Liouville problems with additional transmission conditions at the points of interaction. Moreover we shall establish some spectral properties and find asymptotic behaviour of the eigenvalues of the problem under consideration. Observe that this type of problems is studied in the setting of the direct sum of the Hilbert space. We also construct fundamental solutions and discuss some properties of spectrum.