On approximation of the eigenvalues of perturbed periodic Schrödinger operators

Boulton, Lyonell; Levitan, M. A.

doi:10.1088/1751-8113/40/31/010

Cited by 30 publications

(62 citation statements)

References 13 publications

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“…(see [5][6][7]17]). In all these articles, the idea is to reduce the spectral approximation problems into the estimation of a particular function, related to the distance from the spectrum.…”

Section: Analytical Approachmentioning

confidence: 99%

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Kumar¹

2015

Semigroups, Algebras and Operator Theory

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Normal categories are essentially those arising as the category of principal left [right] ideals of a regular semigroup. These categories have been used in describing the structure of regular semigroups. The structure theory in this context is known as cross connection theory. Several associated categories can be derived from a normal category which are also of interest in the structure theory of regular semigroups. The subcategory of inclusions, the subcategory of retractons, the groupoid of isomorphisms etc. are some of the associated categories.

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Section: Analytical Approachmentioning

confidence: 99%

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Kumar¹

2015

Semigroups, Algebras and Operator Theory

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“…Moreover, if we possess rough a priori certified information about the position of the eigenvalues of A κ , the enclosure can be improved substantially. To be precise [26,28] …”

Section: The Second-order Spectrum and Eigenvalue Approximationmentioning

confidence: 99%

“…As we shall demonstrate in the following, an application of this method leads to sharp eigenvalue bounds for the operator associated with A κ . Recently, the quadratic method was applied successfully to crystalline Schrödinger operators [26], the hydrogenic Dirac operator [27] and models from magnetohydrodynamics [28].…”

Section: Introductionmentioning

confidence: 99%

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

Boulton

Winklmeier

2015

Proc. R. Soc. A.

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We examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. The strategy relies on computing the secondorder spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr-Newman Dirac operator, explicit rates of convergence linked to regularity of the eigenfunctions are established. Numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included.

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“…These approximate computations need some care, in order to avoid the problem of 'Spectral Pollution'. (See Davies & Plum 2004;Boulton & Levitin 2007;Teschl 2008. ) …”

Section: How To Find a B − And B +mentioning

confidence: 99%

Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

2011

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Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

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On approximation of the eigenvalues of perturbed periodic Schrödinger operators

Cited by 30 publications

References 13 publications

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

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