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73
Eigenvalue bracketing for discrete and metric graphs
 J. Math. Anal. Appl
"... Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the m ..."
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Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
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We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
I.: Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
"... Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Eigenvalues In Spectral Gaps Of A Perturbed Periodic Manifold
, 2001
"... We consider a noncompact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is ..."
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We consider a noncompact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp.
Discreteness of the Spectrum for Some Differential Operators With Unbounded Coefficients in R^n
"... We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coeffic ..."
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We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients.
Cardiff University
"... La Escuela Venezolana de Matemáticas es una actividad de los postgrados en matemáticas de las siguientes instituciones: Centro de Estudios Avanzados del Instituto Venezolano de Investigaciones Científicas, Facultad de Ciencias de la Universidad Central de Venezuela, Facultad de Ciencias de la Univer ..."
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La Escuela Venezolana de Matemáticas es una actividad de los postgrados en matemáticas de las siguientes instituciones: Centro de Estudios Avanzados del Instituto Venezolano de Investigaciones Científicas, Facultad de Ciencias de la Universidad Central de Venezuela, Facultad de Ciencias de la Universidad de Los Andes, Universidad Simón Bolívar, Universidad Centro Occidental Lisandro Alvarado y Universidad de Oriente, y se realiza bajo el auspicio de la Asociación Matemática Venezolana. La XX ESCUELA VENEZOLANA DE MATEMÁTICAS recibió financiamiento de la Academia de Ciencias Físicas, Matemáticas y Naturales, la Corporación
On the Accurate Finite Element Solution of a Class of Fourth Order Eigenvalue Problems
, 1995
"... This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation ..."
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This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corner on domain boundaries. Recent computational results of Bjrstad and Tjstheim [4], using a highly accurate spectral LegendreGalerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained...
Error estimates on anisotropic Q1 elements for functions in weighted Sobolev spaces
 Math. Comp
"... Abstract. InthispaperweproveerrorestimatesforapiecewiseQ1average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions. Our error estimates are valid under the condition that neighboring elements have comparable size. This is ..."
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Abstract. InthispaperweproveerrorestimatesforapiecewiseQ1average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions. Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses. Moreover, we generalize the error estimates allowing on the righthand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems. Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements. As an application we consider the approximation of a singularly perturbed reactiondiffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained. 1.
Nonlinear Eigenvalue Problems Of Schrödinger Type Admitting Eigenfunctions With Given Spectral Characteristics
"... The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form A 0 y +B(y)y = y (*) in a real Hilbert space H with a semibounded selfadjoint operator A 0 , while for every y from a dense subspace X of H, B(y) is a symmetric operator. T ..."
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The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form A 0 y +B(y)y = y (*) in a real Hilbert space H with a semibounded selfadjoint operator A 0 , while for every y from a dense subspace X of H, B(y) is a symmetric operator. The lefthand side is assumed to be related to a certain auxiliary functional /, and the associated linear problems A 0 v +B(y)v = v (**) are supposed to have nonempty discrete spectrum (y 2 X).We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (*) on a sphere SR := fy 2 Xj kyk H = Rg whose /value is the nth LjusternikSchnirelman level of /j SR and whose corresponding eigenvalue is the nth eigenvalue of the associated linear problem (**), where R ? 0 and n 2 N are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an nth eigenfunction of a linear problem of the form (**). We discuss applications to elliptic partial differential equations with radial symmetry.
Spectral stability of the Neumann Laplacian
 J. Diff. Eq
"... We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of th ..."
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We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.