Spectral Properties of Random Non-self-adjoint Matrices and Operators

Details Spectral Properties of Random Non-self-adjoint Matrices and Operators Spectral Properties of Random Non-self-adjoint Matrices and Operators

2000-02-19

arxiv.org/abs/math/0002159v1

Mathematics

Spectral Theory

keywords: eigenvalues, spectral instability, matrices, computability, pseudospectrum, Schroedinger operator, Anderson model

Scientific paper

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.

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