Study Of Several Classes Of Differential Operators With Transmission Conditions

This paper mainly studies some properties and conclusions of several classes of difer-ential operators with transmission conditions, which have discontinuity points within theirintervals. Such problems originate from many physical problems. Unlike the classical dif-ferential operators, due to the existence of discontinuity points, we need to add some prop-er conditions which are called transmission conditions. This research focuses on the dis-continuity problem in which the part on both sides of the discontinuity point are relevant.Mukhtarov introduced a special inner product space which relies on transmission conditionsand is proved to be Hilbert space. The main contents of this paper are conducted in this newHilbert space.The first chapter introduces a class of two order diferential operators with transmissionconditions and mixed boundary conditions and a new definition of the inner product, providesthe maximum operator and minimum operator, and proves the self-conjoint of this kind ofoperator and some related properties.In the second chapter, on the basis of the first chapter, we discusses the characteristicvalue problem of the two order diferential operators with transmission conditions and mixedboundary conditions. By means of equivalent transformation of transmission conditions, wemake the calculation more simple. We also calculate the asymptotic estimation formula onboth sides of the discontinuity point in the interval.In the third chapter, a class of two order S-L problems with transmission conditionsand two endpoint singularity are studied. First we discuss this problem in a finite interval I.According to the initial conditions, we discuss two diferent cases. Then the conclusion offinite interval I is further extrapolated to infinite interval and finally we obtain the relationshipbetween the Weyl matrix and the spectral matrix.