Study Several Differential Operators With Discontinuous Points In The Interval

In this paper, we focus on the differential operators with discontinuous int-erior points in the interval. The study contains two aspects: the first onediscusses an issue of high-order differential operators with coupled boundaryconditions and finite general forms of transmission. Transmissions are added,with the reason of numerous discontinuous points, at discontinuous points.We construct a new Hilbert space associated with the transmissions, in whichmaximal and minimal operators, the ones related to transmission conditions,are defined. Applying the general theory of differential operators, we establishthe necessary and sufficient conditions of self-conjugate of the high-orderdifferential operators with coupled boundary conditions and finite generalforms of transmission, furthermore, by using fundamental solutions of thedifferential equation with piecewise coefficients, the problem of eigenvaluesis converted to that of the zeros of the characteristic function, we prove thatzeros of the entire function are the eigenvalues for the problems, and can getnecessary and sufficient conditions of eigenvalues. Thus we give theeigenvalues of the boundary value problems are isolated with no finiteaccumulation point and there are only at most countable number of them;moreover, we construct Green function of the discontinuous differentialoperator, and prove that the compactness of the inverse operator and thecompleteness of their eigenfunctions. The second one discusses problems ofdiscontinuous Sturm-Liouville operator with an indefinite weight functionand both eigenparameter-dependent boundary conditions. Their self-conju-gate, eigenvalues and the Green function are studied. Through the constru-ction of a Krein space related with the boundary value problem, the self-conj- ugate of the new operator can be proved; and the further construction ofcharacterization of the eigenvalues as zeros of an entire function is done aswell as the further construction of Green function. We make a transformationof the problem of eigenvalues into that of the zeros of the entire function, thenobtain the necessary and sufficient conditions of eigenvalues.