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On The Eigenvalues of Some Vectorial Sturm-Liouville Eigenvalue Problems  [PDF]
Hua-Huai Chern
Mathematics , 1999,
Abstract: The author tries to derive the asymptotic expression of the large eigevalues of some vectorial Sturm-Liouville differential equations. A precise description for the formula of the square root of the large eiegnvalues up to the $O(1/n)$-term is obtained.
On the Construction of Isospectral Vectorial Sturm-Liouville Differential Equations  [PDF]
Hua-Huai Chern
Mathematics , 1999,
Abstract: The author extends the idea of Jodeit and Levitan for constructing isospectral problems of the classical scalar Sturm-Liouville differential equations to the vectorial Sturm-Liouville differential equations. Some interesting relations are found.
Inequalities and eigenvalues of Sturm-Liouville problems near a singular boundary
Everitt WN,Marletta M,Zettl A
Journal of Inequalities and Applications , 2001,
Abstract: We study the behavior of eigenvalues of Sturm-Liouville problems (SLP) when an endpoint of the underlying interval approaches a singularity.
Inequalities among eigenvalues of Sturm–Liouville problems  [cached]
Eastham MSP,Kong Q,Wu H,Zettl A
Journal of Inequalities and Applications , 1999,
Abstract: There are well-known inequalities among the eigenvalues of Sturm–Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary conditions. In this paper, for an arbitrary coupled self-adjoint boundary condition, we identify two separated boundary conditions corresponding to the Dirichlet and Neumann conditions in the classical case, and establish analogous inequalities. It is also well-known that the lowest periodic eigenvalue is simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm–Liouville problems with coupled boundary conditions are the same. An important step in our approach is to obtain a representation of the fundamental solutions for sufficiently negative values of the spectral parameter. Our approach yields the existence and boundedness from below of the eigenvalues of arbitrary self-adjoint regular Sturm–Liouville problems without using operator theory.
Dependence of Discrete Sturm-Liouville Eigenvalues on Problems  [PDF]
Hao Zhu,Shurong Sun,Yuming Shi,Hongyou Wu
Mathematics , 2015,
Abstract: This paper is concerned with dependence of discrete Sturm-Liouville eigenvalues on problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.
Eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval  [PDF]
Bilal Chanane
Mathematics , 2006,
Abstract: In this work, we use the \textit{regularized sampling method} to compute the eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval. We work out an example by computing a few eigenvalues and their corresponding eigenfunctions.
Computing eigenvalues of regular Sturm-Liouville problems
H. I. Dwyer,A. Zettl
Electronic Journal of Differential Equations , 1994,
Abstract: of regular self-adjoint Sturm-Liouville problems with matrix coefficients and arbitrary coupled boundary conditions.
Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm--Liouville equations  [PDF]
Nataliya Pronska
Mathematics , 2013,
Abstract: We study asymptotics of eigenvalues, eigenfunctions and norming constants of singular energy-dependent Sturm--Liouville equations with complex-valued potentials. The analysis essentially exploits the integral representation of solutions, which we derive using the connection of the problem under study and a Dirac system of a special form.
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics  [PDF]
Veerle Ledoux,Marnix Van Daele,Guido Vanden Berghe
Mathematics , 2008, DOI: 10.1016/j.cpc.2008.10.001
Abstract: Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.
A priori bounds and existence of non-real eigenvalues of indefinite Sturm-Liouville problems  [PDF]
Jiangang Qi,Shaozhu Chen
Mathematics , 2013,
Abstract: The present paper gives a priori bounds on the possible non-real eigenvalues of regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.
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