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. AMS subject classifications: 65F15, 65F22, 15A18, 15A52, 47A10, 47A75, 47B80, 60H25. keywords: eigenvalues, spectral instability, matrices, computability, pseudospectrum, Schrödinger operator, Anderson model.