1,127 research outputs found
Approximated structured pseudospectra
Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra
Pseudospectra in non-Hermitian quantum mechanics
We propose giving the mathematical concept of the pseudospectrum a central
role in quantum mechanics with non-Hermitian operators. We relate
pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint
operators, and basis properties of eigenfunctions. The abstract results are
illustrated by unexpected wild properties of operators familiar from
PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion
excluding basis property (Proposition 6) added, unbounded time-evolution
discussed, new reference
On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators
In this paper we develop and apply methods for the spectral analysis of
non-self-adjoint tridiagonal infinite and finite random matrices, and for the
spectral analysis of analogous deterministic matrices which are pseudo-ergodic
in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a
major application to illustrate our methods we focus on the "hopping sign
model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443),
in which the main objects of study are random tridiagonal matrices which have
zeros on the main diagonal and random 's as the other entries. We
explore the relationship between spectral sets in the finite and infinite
matrix cases, and between the semi-infinite and bi-infinite matrix cases, for
example showing that the numerical range and -norm \eps-pseudospectra
(\eps>0, ) of the random finite matrices converge almost
surely to their infinite matrix counterparts, and that the finite matrix
spectra are contained in the infinite matrix spectrum . We also propose
a sequence of inclusion sets for which we show is convergent to
, with the th element of the sequence computable by calculating
smallest singular values of (large numbers of) matrices. We propose
similar convergent approximations for the 2-norm \eps-pseudospectra of the
infinite random matrices, these approximations sandwiching the infinite matrix
pseudospectra from above and below
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