Spectral Problems Of Several Classes Of Operators

With development of applications in other disciplines and many engineering technologies,the study of spectral problem of operators has aroused great interest and attention to domestic and overseas scholars.So far,the spectral problems of matrix operators and some related Laplace operators have become two topics that have developed and grown rapidly in operator theory.They have wide and direct applications in the fields of geophysics,quantum physics,chemistry and chemical engineering,etc.,and they are also effective ways to solve equations in mathmatics and mathematical physics.In this thesis,first we investigate the spectral problem of quasi-tridiagonal matrix operators.The spectral problem of matrix operators has been one of the most active research topics in spectral problem of operators in recent years.At the same time,the study of the spectral problem of quasi-tridiagonal matrix operators can be applied to study dissipative operators in infinite dimensional spaces.Then the spectral problems of Laplace-Beltrami operator and fractional Laplace operator are considered,respectively.The main works are as follows:The spectral problems of quasi tridiagonal matrix operators are studied.Based on the study of the spectral properties of quasi tridiagonal matrix operators,the inverse eigenvalue problem of quasi tridiagonal matrix is reconstructed according to the eigenvalue data satisfying the specific spectral properties,and the sufficient conditions for the existence of the solutions of the inverse eigenvalue problem of quasi tridiagonal matrix are obtained.This is about solvability.The algorithm of constructing this kind of matrix is obtained by unitary matrix tool.This is about computability.Finally,several examples are given to illustrate the feasibility of the algorithm.In particular,we analyze the spectral properties of quasi tridiagonal matrix from two aspects which quasi tridiagonal matrix and its main submatrix have or do not have common eigenvalues to characterize the eigenvalues including multiplicity and location specifically.We obtain approximation bounds for products of quasimodes for the LaplaceBeltrami operator on compact Riemannian manifolds of all dimensions without boundary.The products of quasimodes can be approximated by a low-degree vector space,and the size of the space is small.Based on bilinear quasimode estimates of all dimensions,we give approximation bounds in Coulomb norm.We also prove approximation bounds forthe products of quasimodes in 2-norm using the results of quasimodes.In particular,the bilinear quasimode estimates are studied from two aspects of lower dimension and higher dimension,respectively,and bilinear quasimode estimates on compact Riemannian manifolds of all dimensions without boundary are obtained,to make the highest frequency disappear from the estimates.Furthermore,the result of the case where the eigenvalues corresponding to bilinear quasimodes are equal to bilinear quasimode estimates improves part of quasimodes estimates of Sogge-Zelditch in eight dimensions and above.The conclusion of this part is to extend the results of eigenfunctions to quasimodes.The asymptotic formula of the eigenvalues for the spectral problem of onedimensional fractional Laplace operator in the interval is derived.We obtain results of the asymptotic law for the eigenvalues.That is,the estimate of eigenvalues is valid when parameter in fractional Laplace operator approaches to 0.The asymptotic formula(without the information about the order of the error term)of the eigenvalues of the spectral problems of fractional Laplace operators in this part can be verified by numerical experiments in physics.Furthermore,based on the results of the asymptotic distribution of eigenvalues,the property of eigenfunction is studied.We give the approximation to the eigenfunction.