2000/02/19 by E. B. Davies, E B Davies, Davies, E B
Mathematics · Physics and Astronomy · #15A18 #15A52 #47A75 #47B80 #60H25 #65F15 #65F22 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.SP #msc:15A18 #msc:15A52 #msc:47A75 #msc:47B80 #msc:60H25 #msc:65F15 #msc:65F22
paper · pdf · doi:10.48550/arxiv.math/0002159
keywords: eigenvalues, spectral instability, matrices, computability, pseudospectrum, Schroedinger operator, Anderson model
arxiv created 2000/02/19 · openalex publication_date 2000/02/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.