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On the first two eigenvalues of the Sturm-Liouville operatorsMasehla, Johannes Namo 06 August 2008 (has links)
No description available.
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Generalized Sturm-Liouville theory for dissipative systems. / 耗散系統中的廣義Sturm-Liouville理論 / Generalized Sturm-Liouville theory for dissipative systems. / Hao san xi tong zhong de guang yi Sturm-Liouville li lunJanuary 2004 (has links)
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
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The nature of spectrum for some singular Sturm-Liouville operatorsLee, Shuo-Chi 23 July 2006 (has links)
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.
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The application of asymptotic forms to an expansion problem of the Sturm Liouville type where the coefficient of the parameter changes signBarron, James Joseph. January 1934 (has links)
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.
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Spektrum und asymptotische Eigenwertverteilung singulärer Sturm-Liouville-Probleme mit indefiniter GewichtsfunktionSchroeder, Martin. January 1997 (has links)
Duisburg, Universiẗat, Diss., 1997. / Dateiformat: zip, Dateien in unterschiedlichen Formaten.
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Sturmian theory and its applicationsLawson, R. D. Unknown Date (has links)
No description available.
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Optimal upper bounds of eigenvalue ratios for the p-LaplacianChen, Chao-Zhong 19 August 2008 (has links)
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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Boundary and internal layers in a semilinear parabolic problemSalazar-González, José Domingo 05 1900 (has links)
No description available.
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Cálculo de funções de Green pelo método de expansão tipo Sturm-LiouvilleOliveira, Edmundo Capelas de, 1952- 21 July 1979 (has links)
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
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Spectral theory of differential operators on graphsCurrie, Sonja 31 October 2006 (has links)
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.
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