Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition

In this paper we obtain asymptotic estimates of eigenvalues for regular Sturm-Liouville problems having the eigenparameter in the boundary condition. The method is based on an iterative procedure solving the associated Riccati equation and producing an asymptotic expansion of the solution in the higher powers of 1/λ1/2 as λ→∞.     

The sampling method for sturm-liouville problems with the eigenvalue parameter in the boundary condition.

ABSTRACT. We shall apply the sampling method to compute the eigenvalues of a regular Sturm Liouville problem when a boundary condition contains the eigenvalue parameter. Truncation errors will be worked out and help us obtain eigenvalue enclosures. Examples are provided to illustrate the theory.     

Regular multiparameter eigenvalue problems with several parameters in the boundary conditions

This paper extends the work of J. Walter on regular eigenvalue problems with eigenvalue parameter in the boundary condition to the multiparameter setting. The main results include several completeness and expansion theorems.     

The first Robin eigenvalue with negative boundary parameter

We give a counterexample to the long standing conjecture that the ball maximises the first eigenvalue of the Robin eigenvalue problem with negative parameter among domains of the same volume. Furthermore, we show that the conjecture holds in two dimensions provided that the boundary parameter is small. This is the first known example within the class of isoperimetric spectral problems for the first eigenvalue of the Laplacian where the ball is not an optimiser.     

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on and Sturm-Liouville problems with -holomorphic boundary conditions are considered.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A LOWER BOUND FOR FIRST EIGENVALUE WITH MIXED BOUNDARY CONDITIOIN

Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature RicM≥n- 1. The paper obtains an inequality for the first eigenvalue η1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz,Reilly, Escobar and Xia. It is also proved that η1≥ n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.     

On condition numbers of polynomial eigenvalue problems

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.     

Regular boundary value problems with a discontinuous coefficient,functional-multipoint conditions,and a linear spectral parameter

We consider a linear differential equation which includes a linear abstract operator, with a piecewise constant coefficient at the principal differential term, together with multipoint boundary-transmission conditions, including linear functionals. The spectral parameter appears linearly in the equation and may appear also linearly in the boundary-transmission conditions. We prove an isomorphism and coerciveness of the problem with respect to the spectral parameter and the space variable.     

Discrete Sturm-Liouville problems with nonlinear parameter in the boundary conditions

This paper deals with discrete second order Sturm-Liouville problems where the parameter that is part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter

A differential eigenvalue problem with a nonlinear dependence on the parameter is approximated by the finite-element method with numerical integration. We study the error in the approximate eigenvalues and eigenfunctions.     

Transmission eigenvalue problem for inhomogeneous absorbing media with mixed boundary condition

Consider the transmission eigenvalue problem for the wave scattering by a dielectric inhomogeneous absorbing obstacle lying on a perfect conducting surface. After excluding the purely imaginary transmission eigenvalues, we prove that the transmission eigenvalues exist and form a discrete set for inhomogeneous non-absorbing media, by using analytic Fredholm theory. Moreover, we derive the Faber-Krahn type inequalities revealing the lower bounds on real transmission eigenvalues in terms of the media parameters. Then, for inhomogeneous media with small absorption, we prove that the transmission eigenvalues also exist and form a discrete set by using perturbation theory. Finally, for homogeneous media, we present possible components of the eigenvalue-free zone quantitatively, giving the geometric understanding on this problem.     

On componentwise condition numbers for eigenvalue problems with structured matrices

We give explicit expressions for the componentwise condition number for eigenvalue problems with structured matrices. We will consider only linear structures and show a general result from which expressions for the condition numbers follow. We obtain explicit expressions for the following structures: Toeplitz and Hankel. Details for other linear structures should follow in a straightforward manner from our general result. Copyright © 2008 John Wiley & Sons, Ltd.     

Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition

We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra.     

Spectra of discrete symplectic eigenvalue problems with separated boundary conditions

In this paper we consider a discrete symplectic eigenvalue problem with separated boundary conditions and obtain formulas for the number of eigenvalues on a given interval of the variation of the spectral parameter. In addition, we compare the spectra of two symplectic eigenvalue problems with different separated boundary conditions.