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Lau Ching Yan Ada = 耗散系統中的廣義Sturm-Liouville理論 / 劉正欣. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 156-157). / Text in English; abstracts in English and Chinese. / Lau Ching Yan Ada = Hao san xi tong zhong de guang yi Sturm-Liouville li lun / Liu Zhengxin. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Vibrational motion in physics --- p.1 / Chapter 1.2 --- Normal modes of vibration --- p.2 / Chapter 1.3 --- Boundary conditions --- p.4 / Chapter 1.4 --- The wave equation --- p.6 / Chapter 1.4.1 --- Mechanical waves --- p.7 / Chapter 1.4.2 --- Electromagnetic waves --- p.9 / Chapter 1.5 --- General form of the wave equation --- p.10 / Chapter 1.5.1 --- V(x) as a restoring force --- p.11 / Chapter 1.5.2 --- V(x) in gravitational waves --- p.13 / Chapter 1.5.3 --- V(x) by transformation --- p.16 / Chapter 2 --- Sturm-Liouville systems --- p.18 / Chapter 2.1 --- Introduction --- p.18 / Chapter 2.2 --- Differential operators --- p.19 / Chapter 2.2.1 --- Introduction --- p.19 / Chapter 2.2.2 --- Adjoint operators --- p.20 / Chapter 2.2.3 --- Self-adjoint operators --- p.21 / Chapter 2.2.4 --- More examples --- p.24 / Chapter 2.3 --- Sturm-Liouville boundary-value problems --- p.27 / Chapter 2.4 --- Sturm-Liouville theory --- p.28 / Chapter 2.4.1 --- Real eigenvalues --- p.29 / Chapter 2.4.2 --- Orthogonal eigenfunctions --- p.30 / Chapter 2.4.3 --- Completeness of eigenfunctions --- p.31 / Chapter 2.4.4 --- Interlacing zeros of the eigenfunctions --- p.33 / Chapter 2.5 --- Applications of Sturm-Liouville theory --- p.35 / Chapter 2.5.1 --- Vibrations of a string --- p.36 / Chapter 2.5.2 --- The hydrogen atom --- p.40 / Chapter 3 --- Wave equation with damping --- p.46 / Chapter 3.1 --- Statement of problem --- p.46 / Chapter 3.1.1 --- The equation --- p.46 / Chapter 3.1.2 --- The operator --- p.48 / Chapter 3.1.3 --- Non-self-adjointness --- p.49 / Chapter 3.2 --- Eigenfunctions and Eigenvalues --- p.51 / Chapter 3.3 --- The completeness problem --- p.53 / Chapter 4 --- Green's function solution --- p.55 / Chapter 4.1 --- Introduction --- p.55 / Chapter 4.2 --- Green's function solution --- p.56 / Chapter 4.3 --- Fourier transform --- p.58 / Chapter 4.4 --- Inverse Fourier transform --- p.61 / Chapter 5 --- Proof of completeness --- p.66 / Chapter 5.1 --- WKB approximation --- p.66 / Chapter 5.2 --- "An upper bound for \G(x,y,w)e~iwt\ " --- p.68 / Chapter 5.3 --- Proof of completeness --- p.72 / Chapter 5.3.1 --- The limit when R→∞ --- p.72 / Chapter 5.3.2 --- Eigenfunction expansion --- p.76 / Chapter 6 --- The bilinear map --- p.80 / Chapter 6.1 --- Introduction --- p.80 / Chapter 6.2 --- Evaluation of J1(wj) --- p.82 / Chapter 6.3 --- Self-adjointness of H --- p.84 / Chapter 6.4 --- Properties of the map --- p.87 / Chapter 7 --- Applications --- p.89 / Chapter 7.1 --- Eigenfunction expansion --- p.89 / Chapter 7.2 --- Perturbation theory --- p.94 / Chapter 7.2.1 --- First and second-order corrections --- p.95 / Chapter 7.2.2 --- Example --- p.97 / Chapter 7.2.3 --- Example (Constant r) --- p.102 / Chapter 8 --- Critical points --- p.104 / Chapter 8.1 --- Introduction --- p.104 / Chapter 8.2 --- Conservative cases (Γ = 0) --- p.105 / Chapter 8.3 --- Non-conservative cases (Constant r) --- p.107 / Chapter 8.4 --- Critical points away from imaginary axis --- p.108 / Chapter 9 --- Jordan block and applications --- p.114 / Chapter 9.1 --- Jordan basis --- p.114 / Chapter 9.2 --- An analytical example --- p.117 / Chapter 9.2.1 --- Solving for the extra basis function --- p.117 / Chapter 9.2.2 --- Freedom of choice --- p.118 / Chapter 9.2.3 --- Interpolating function --- p.120 / Chapter 9.3 --- A numerical example --- p.122 / Chapter 9.3.1 --- "Solving for f2,1 " --- p.124 / Chapter 9.3.2 --- Interpolating function --- p.126 / Chapter 9.4 --- Jordan basis expansion --- p.127 / Chapter 9.5 --- Perturbation theory near critical points --- p.131 / Appendices --- p.142 / Chapter A --- WKB approximation --- p.142 / Chapter B --- Green's function (Discontinuous V(x)) --- p.145 / Chapter B.l --- Finite discontinuouity in V(x) --- p.145 / Chapter B.1.1 --- Green's function --- p.145 / Chapter B.1.2 --- "Behaviour of the extra phases Φ, Φ " --- p.147 / Chapter B.2 --- Delta function in --- p.148 / Chapter B.2.1 --- Green's function --- p.148 / Chapter B.2.2 --- "Behaviour of the extra phases Φ, Φ " --- p.150 / Chapter C --- Dual basis --- p.151 / Chapter C.1 --- Matrix representation --- p.152 / Chapter C.2 --- Relation with bilinear map --- p.153 / Chapter C.3 --- Construction of dual basis --- p.154 / Bibliography --- p.156

We give a report on the Sturm-Liouville problem defined on semi-infinite interval. Here as an extension of the Fourier expansion, we have a Parseval equality involving a Fourier integral with respect to a spectral function rho. This function rho is also related to Titchmarsh-Weyl m-function m(lambda) giving L2 solutions of the problem. The spectrum can be viewed as nonconstant points of the spectral function. Following Titchmarsh¡¦s monograph, we shall investigate the nature of the spectrum associated with different asymptotic behaviors of the potential function q, namely, when q¡÷¡Û, q¡÷0 or q¡÷-¡Û.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1934. / Typescript and manuscript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Duisburg, Universiẗat, Diss., 1997. / Dateiformat: zip, Dateien in unterschiedlichen Formaten.

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In this thesis, we study the optimal estimate of eigenvalue ratios £f_n/£f_m of the Sturm-Liouville equation with Dirichlet boundary conditions on (0, £k). In 2005, Horvath and Kiss [10] showed that £f_n/£f_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that £f_n/£f_1≤n^2 when q ≥ 0. Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(£s£c(x)), y'(x)=r(x)£scos(£s£c(x)), where £s = ¡Ô£f. This modified phase seems to be more effective than the phases £p and £r that Horvath and Kiss [10] used. Furthermore our approach can be generalized to study the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet problem of the equation -[(y')^(p-1)]'=(p-1)(£f-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies £f_n/£f_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, £k_p). Again this is an optimal upper estimate.

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Orientador: Jose Bellandi Filho / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-07-15T03:55:20Z (GMT). No. of bitstreams: 1 Oliveira_EdmundoCapelasde_M.pdf: 755219 bytes, checksum: 80ba55e1d5a784ef3a819212b87f373f (MD5) Previous issue date: 1979 / Resumo: Não informado / Abstract: Not informed. / Mestrado / Física / Mestre em Física

Student Number : 9804032J - PhD thesis - School of Mathematics - Faculty of Science / The focus of this thesis is the spectral structure of second order self-adjoint differential operators on graphs. Various function spaces on graphs are defined and we define, in terms of both differential systems and the afore noted function spaces, boundary value problems on graphs. A boundary value problem on a graph is shown to be spectrally equivalent to a system with separated boundary conditions. An example is provided to illustrate the fact that, for Sturm-Liouville operators on graphs, self-adjointness does not necessarily imply regularity. We also show that since the differential operators considered are self-adjoint the algebraic and geometric eigenvalue multiplicities are equal. Asymptotic bounds for the eigenvalues are found using matrix Pr¨ufer angle methods. Techniques common in the area of elliptic partial differential equations are used to give a variational formulation for boundary value problems on graphs. This enables us to formulate an analogue of Dirichlet-Neumann bracketing for boundary value problems on graphs as well as to establish a min-max principle. This eigenvalue bracketing gives rise to eigenvalue asymptotics and consequently eigenfunction asymptotics. Asymptotic approximations to the Green’s functions of Sturm-Liouville boundary value problems on graphs are obtained. These approximations are used to study the regularized trace of the differential operators associated with these boundary value problems. Inverse spectral problems for Sturm-Liouville boundary value problems on graphs resembling those considered in Halberg and Kramer, A generalization of the trace concept, Duke Math. J. 27 (1960), 607-617, for Sturm-Liouville problems, and Pielichowski, An inverse spectral problem for linear elliptic differential operators, Universitatis Iagellonicae Acta Mathematica XXVII (1988), 239-246, for elliptic boundary value problems, are solved. Boundary estimates for solutions of non-homogeneous boundary value problems on graphs are given. In particular, bounds for the norms of the boundary values of solutions to the non-homogeneous boundary value problem in terms of the norm of the non-homogeneity are obtained and the eigenparameter dependence of these bounds is studied. Inverse nodal problems on graphs are then considered. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph from which the uniqueness of the potential, for given nodal data, is deduced. An explicit formula for the potential in terms of the nodal points and eigenvalues is given.