Spectral Analysis And Numerical Computation Of Sturm-Liouville Problems

In this paper,we study three problems in the field of ordinary differential operators:relations among spectrum of different types of differential operators,numerical computation and the spectral analysis of differential operators with transmission conditions.Relations among spectrum of different types of differential operators are important issues in the Sturm-Liouville(S-L) theory.For any S-L problem with a separable boundary condition and whose leading coefficient function changes sign,we first give a geometric characterization of its eigenvaluesλ_n using the eigenvalues of some corresponding problems with a definite leading coefficient function.Consequences of this characterization include simple proofs of the existence of theλ_n's, their Prüfer angle characterization,and a way for determining their indices from the zeros of their eigenfunctions.Then,interlacing relations among theλ_n's and the eigenvalues of the corresponding problems are obtained.Using these relations,we give a simple proof of asymptotic formulas for theλ_n's.To the best of our knowledge,these relations are new.Such relations give a new way and a new geometric idea for studying the spectrum of relatively complicated S-L problems using the spectrum of simpler problems.S-L problems with transmission conditions,problems with spectral parameter dependent boundary conditions and transmission conditions and non-self-adjoint S-L problems are new issues in the spectrum theory of ordinary differential operators.Applying the property of the characteristic function,the computation of the eigenvalues of S-L problems is converted to that of the zeros of the characteristic function.When the characteristic function is not explicitly available,the root isolation is not easy.We give the algorithms for computing the eigenvalues,to- gether with their multiplicities,and the corresponding eigenfunctions. The algorithms,whose structure is simple and whose thinking is clear, provide a uniform frame of solving S-L problems.The algorithms can approximate either real or non-real eigenvalues.For the convenience of numerical simulations,a general method of constructing examples with known eigenvalues and eigenfunctions is given,and the method,which react on studying the oscillation of eigenfunctions of S-L problems with transmission conditions,shows the originality.The other task:Using some theoretical results about the index problem,the computation of the indices of known eigenvalues of self-adjoint S-L problems with coupled boundary conditions is converted to that of the indices of the same eigenvalues for appropriate separated boundary conditions,and is then carried out in terms of the Prufer angle.The algorithm so generated is discussed,and numerous examples are presented to illustrate the theoretical results and the practicability and effectiveness of algorithms mentioned above.We investigate S-L problems with transmission conditions at an interior point,including the reality of eigenvalues and the oscillation of eigenfunctions.We Analyze numerical examples and graphs of eigenfunctions obtained by our algorithms,and relate the oscillation of eigenfunctions corresponding toλ_n to the relation between "jump state" of the eigenfunction corresponding toλ₁ with "jump state" of the eigenfunction corresponding toλ_n in the interior point,then yield the oscillatory behavior via directly surveying examples and then prove the oscillation of eigenfunctions.The way of studying oscillatory properties of eigenfunctions by numerical simulations is new.For S-L problems with spectral parameter dependent boundary conditions and transmission conditions,the fact that operator is self-adjoint in an appropriate space is proved and the properties of its eigenvalues are discussed,establishing a new operator associated with the problems.This paper contains seven chapters.The first chapter:the background of the problems investigated and main results obtained in this paper.The second chapter:eigenvalue inequalities among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions.The third chapter:numerical computations of the regular continuous Sturm-Liouville problems.The fourth chapter:numerical computations of the indices of Sturm-Liouville problems for coupled boundary conditions.The fifth chapter:ordinary differential operators with spectral parameter dependent boundary conditions and transmission conditions.The sixth chapter:numerical computations of the eigenvalues and eigenfunctions of regular Sturm-Liouville problems with transmission conditions.The last chapter:oscillatory properties of eigenfunctions of Sturm-Liouville problems with transmission conditions.