The Finite Spectrum Of Differential Boundary Value Problems And Its Matrix Representations

In this paper, we investigate the differential boundary value prob-lems with a finite spectrum (including fourth order problems;2nth order problems; Sturm-Liouville problems with transmission conditions; Sturm-Liouville problems with transmission conditions and parameter-dependent boundary conditions) and their corresponding equivalence to finite dimen-sional matrix eigenvalue problems.In1964, Atkinson in his well known book "Discrete and Continuous Boundary Value Problems"[2] suggested that for the second order Sturm-Liouville problems consisting with the equation-(py′)′+qy=λwy(on J=(a,b), with-∞＜a＜b＜+∞), under some conditions, may have a finite spectrum. But, he did not elaborate with either an example or a theorem. In order to confirm Atkinson's this argument is reasonable, in2001Kong, Wu and Zettl [40] constructed a class of Sturm-Liouville prob-lems which consist of a finite number of eigenvalues. They show that, for any positive integer m, there exists a class of second order Sturm-Liouville problems which have exactly m eigenvalues, and further they show that these m eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case. In2009, Kong, Volkmer, Zettl [39] give these problems a matrix rep-resent at ions (i.e. the equivalence between the second order Sturm-Liouville problems with finite spectrum and the finite dimensional matrix eigenvalue problems). This illustrated that under some special conditions the two prob- lems can be "transferred" from one to the other. This will enrich the study of theoretical approaches for both problems.In this paper, based on the results mentioned above, we extend the finite spectrum results and corresponding matrix representations to fourth order cases, even2nth order cases and the Sturm-Liouville problems with transmission conditions or eigenparameter-dependent boundary conditions.Firstly, we consider the finite spectrum of fourth order boundary value problems. Through the special partition on the interval and the coefficients of each subinterval to meet certain special conditions, we obtain the itera-tive formula of the characteristic function, then the characteristic function on the parameters as a finite polynomial of λ is derived. According to prop-erty that the zeros of the characteristic function are the eigenvalues of the problem itself, we can come to the conclusion that there is a finite num-ber of eigenvalues. Due to the increase of the order, the problem becomes quite complex. Through a lot of analysis and deduction, the conditions of fourth order problems with finite spectrum results and the iteration formula of the characteristic function of fourth order boundary value problems are given. This is the key to solving this problem; Further, we investigated the2nth order boundary value problems with finite spectrum, and extended the iteration formula of the characteristic function from the perspective of mathematical induction, and we also find the relationship between the maximum of the eigenvalue numbers and the order of the equations and the number of the partition of the interval.Both the differential boundary value problems and matrix eigenvalue problems have their own practical application backgrounds, therefore a clear understanding of the relationship between the two problems in certain spe-cial circumstances, whether in theory or in applications, are of great signif- icance. Hence, we study the equivalence between the fourth order bound-ary value problems with finite spectrum and the finite-dimensional matrix eigenvalue problems. By considering a class of fourth order boundary value problems which can be called as Atkinson type, we give the matrix repre-sentations of it. Also due to the increase of the order, the matrix form has become complex. To this end, we introduce block matrix concept, and get that the matrix representations of fourth order boundary value problems are with tridiagonal form in the sense of block matrix literature. Not only that, the matrix also has the good characteristics of the symmetrical form.At the same time, the paper also studied a class of differential opera-tors with "discontinuity", i.e. Sturm-Liouville problems with transmission conditions at an interior point, which are concerned by many mathematical and physical researchers. We considered a class of Sturm-Liouville prob-lems with transmission conditions and their matrix representations. For this problem, we introduce the concept of correlation matrix of transmis-sion conditions, and the iterative formula of characteristic function with transmission conditions are given, and then the finite eigenvalue results are obtained. Moreover, since the transmission conditions being appeared, the number of eigenvalues will be affected, hence we show the conclusion of the finite spectrum and the relationship with the transmission conditions through a example; In accordance with the corresponding matrix represen-tations, we considered not only the separated boundary conditions, but also the coupled boundary conditions, and the matrix forms are no longer with Jacobi type or cyclic Jacobi type, hence the'almost Jacobi'matrix has been introduced, which is a class of generalized Jacobi matrices.Finally, the paper also examined a class of Sturm-Liouville problems with spectral parameters in the boundary conditions, and considered the finite spectrum and matrix representation of it. The eigenvalue numbers of such problems will also be subject to the influence of the boundary con-ditions of the spectral parameters, and the matrix forms in the matrix representations are of a class of generalized Jacobi matrices.This paper contains eight parts.1. The background and the main results in this paper;2. The fourth order boundary value problems with finite spectrum;3. Matrix representations of fourth order boundary value problems with finite spectrum;4.2nth order boundary value problems with finite spectrum;5. The finite spectrum of Sturm-Liouville problems with transmission conditions;6. Matrix representations of Sturm-Liouville problems with transmission conditions with finite spectrum;7. The fi-nite spectrum of Sturm-Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions;8. Matrix representations of Sturm-Liouville problems with eigenparameter-dependent boundary con-ditions with finite spectrum.