Criteria For The Discrete Spectrum Of Several Classes J-self-adjoint Differential Operator

Differential operator spectrum theory is one of the basic problems ofdifferential operator theory, it includes the qualitative analysis of spectrumof differential operator, the asymptotic estimates, expansion according tothe eigenfunction, etc. Because it is closely related with the application, especially many quantum mechanics problems resolved by using spectrum analysis of singular differential operator, so it attract widespread attention ofresearchers in different fields. The research work has never stop on spectrum analysis of differential operator, especially, domestic and foreign mathematicians got a series of important results in the qualitative analysis of thespectrum, and the results are mostly concentrated in the self-adjoint differrential operator, but the results about J-self-adjoint differential operatorsare rare.In this paper，we mainly study qualitative analysis of spectrum of differrrential operators and asymptotic estimates. First of all, we analysis andstudy the discreteness of spectrum of several classes of J-self-adjoint differrential operator by using the method of direct sum decomposition and quadratic comparation, finally asymptotic estimates of the eigenvalues of a classof Sturm-Liouville operator is studied.This paper contains six parts. The frist part: an introduction of the bac kground of the problems we study and the progress of the research work,and the necessary fundamental lemma, symbols and other related knowledge we need. The second part: we discuss the discrete spectrum of a classof J-self-adjoint Euler differential operators by using the method of quadratic form comparison, complete continuity of resolvent operator and the method of direct sum decomposition, Some criterion for the discreteness of spectrum are obtained. The third part: The discreteness of spectrum of J-self-adjoint differential operators with exponential coefficients is discussed byusing the method of direct sum decomposition and quadratic form comparison, Some sufficient condition for the discreteness of spectrum of thedifferential operators are obtained when the coefficients satisfying certainconditions. The fourth part: The discreteness of spectrum of J-self-adjointdifferential operators with exponential and power functions coefficients isdiscussed by using the method of direct sum decomposition and quadraticform comparison, Some sufficient condition for the discreteness of spectrum of the differential operators are obtained when the coefficients satisfyying certain conditions. The fifth part: The discreteness of spectrum of aclass of J-self-adjoint differential operators is discussed by using operatordecomposition methods and the real and imaginary parts of open ideas, wefind that the discreteness of the spectrum of such differential operators notonly determined by the last term coefficient which tends to infinity according to a certain way, Moreover, the middle term coefficient which tends to infinity according to a certain way also can decide the discreteness of thespectrum. The sixth part: The asymptotic estimates of eigenvalues for a regular Strum-Liouville problem of finite interval is considered by using thesame order infinitesimal comparative method, We observe that the eigenvalues depend on the coefficient of differential equation and boundary condition, and also give the error is smaller, finer asymptotic results.