557 research outputs found
Spectral Properties of Random Non-self-adjoint Matrices and Operators
We describe some numerical experiments which determine the degree of spectral
instability of medium size randomly generated matrices which are far from
self-adjoint. The conclusion is that the eigenvalues are likely to be
intrinsically uncomputable for similar matrices of a larger size. We also
describe a stochastic family of bounded operators in infinite dimensions for
almost all of which the eigenvectors generate a dense linear subspace, but the
eigenvalues do not determine the spectrum. Our results imply that the spectrum
of the non-self-adjoint Anderson model changes suddenly as one passes to the
infinite volume limit.Comment: keywords: eigenvalues, spectral instability, matrices, computability,
pseudospectrum, Schroedinger operator, Anderson mode
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
On Temple--Kato like inequalities and applications
We give both lower and upper estimates for eigenvalues of unbounded positive
definite operators in an arbitrary Hilbert space. We show scaling robust
relative eigenvalue estimates for these operators in analogy to such estimates
of current interest in Numerical Linear Algebra. Only simple matrix theoretic
tools like Schur complements have been used. As prototypes for the strength of
our method we discuss a singularly perturbed Schroedinger operator and study
convergence estimates for finite element approximations. The estimates can be
viewed as a natural quadratic form version of the celebrated Temple--Kato
inequality.Comment: submitted to SIAM Journal on Numerical Analysis (a major revision of
the paper
Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models
We examine perturbations of eigenvalues and resonances for a class of
multi-channel quantum mechanical model-Hamiltonians describing a particle
interacting with a localized spin in dimension . We consider
unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of
the continuous spectrum and we analyze the effect of various type of
perturbations on the spectral singularities. We provide algorithms to obtain
convergent series expansions for the coordinates of the singularities.Comment: 20 page
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