149 research outputs found
Level sets of the resolvent norm of a linear operator revisited
It is proved that the resolvent norm of an operator with a compact resolvent
on a Banach space cannot be constant on an open set if the underlying space
or its dual is complex strictly convex. It is also shown that this is not the
case for an arbitrary Banach space: there exists a separable, reflexive space
and an unbounded, densely defined operator acting in with a compact
resolvent whose norm is constant in a neighbourhood of zero; moreover is
isometric to a Hilbert space on a subspace of co-dimension . There is also a
bounded linear operator acting on the same space whose resolvent norm is
constant in a neighbourhood of zero. It is shown that similar examples cannot
exist in the co-dimension case.Comment: Final versio
On the convergence of second order spectra and multiplicity
Let A be a self-adjoint operator acting on a Hilbert space. The notion of
second order spectrum of A relative to a given finite-dimensional subspace L
has been studied recently in connection with the phenomenon of spectral
pollution in the Galerkin method. We establish in this paper a general
framework allowing us to determine how the second order spectrum encodes
precise information about the multiplicity of the isolated eigenvalues of A.
Our theoretical findings are supported by various numerical experiments on the
computation of inclusions for eigenvalues of benchmark differential operators
via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper
Sharp eigenvalue enclosures for the perturbed angular Kerr-Newman Dirac operator
A certified strategy for determining sharp intervals of enclosure for the
eigenvalues of matrix differential operators with singular coefficients is
examined. The strategy relies on computing the second order spectrum relative
to subspaces of continuous piecewise linear functions. For smooth perturbations
of the angular Kerr-Newman Dirac operator, explicit rates of convergence due to
regularity of the eigenfunctions are established. Existing benchmarks are
validated and sharpened by several orders of magnitude in the unperturbed
setting.Comment: 27 pages, 2 figures, 5 tables. Some errors fixe
An estimate for the Morse index of a Stokes wave
Stokes waves are steady periodic water waves on the free surface of an
infinitely deep irrotational two dimensional flow under gravity without surface
tension. They can be described in terms of solutions of the Euler-Lagrange
equation of a certain functional. This allows one to define the Morse index of
a Stokes wave. It is well known that if the Morse indices of the elements of a
set of non-singular Stokes waves are bounded, then none of them is close to a
singular one. The paper presents a quantitative variant of this result.Comment: This version contains an additional reference and some minor change
Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II
In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p((.)) -> q((.))-theorems were proved for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x(0) and to infinity, under an additional condition relating the weight exponents at x(0) and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.) (S-n, p) on the unit sphere S-n in Rn+1 are also improved in the same way. (c) 2006 Elsevier Inc. All rights reserved.info:eu-repo/semantics/publishedVersio
On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators
This paper addresses the problem of computing the eigenvalues lying in the
gaps of the essential spectrum of a periodic Schrodinger operator perturbed by
a fast decreasing potential. We use a recently developed technique, the so
called quadratic projection method, in order to achieve convergence free from
spectral pollution. We describe the theoretical foundations of the method in
detail, and illustrate its effectiveness by several examples.Comment: 17 pages, 2 tables and 2 figure
Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients
This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK
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