Analytical resolvent matrix elements and
approximation of pseudospectra
Jean-Philippe Brunet 1
Given an arbitrary discretized operator, or matrix A, we consider the matrix element of the resolvent in the complex z-plane g(z) =< uj(A ? zI)?1ju >, where u is a vector with random components. Since k A ? zI k>= g(z), isocontours of jg(z)j can be used as lower bounds for the boundaries of the ffl-pseudospectra of A. Unlike the norm of the resolvent, g(z) can be expressed as an analytical function of z and can be subsequently displayed at little cost in any region of the complex plane with arbitrary high resolution. Furthermore, our method uses the Arnoldi algorithm and can thus take advantage of sparse representations of operators.
The spectrum of a discretized operator, or a matrix A, is the set of points in the complex z-plane where the resolvent (A ? zI)?1 is singular, hence its norm infinite. What happens in the neighborhhood of those singularities depend on the nature of A. If A is Hermitian (or, strictly speaking, normal, i.e. such as AHA = AAH , where AH is the Hermitian conjugate of A) the norm of the resolvent is large only in the close vicinity of singularities. For instance bounded quantum physical systems, which are described by Hermitian Hamiltonian operators, are known to have a well-defined set of resonnances and indeed atomic frequencies can be determined experimentally with very high precision. When A is non normal the picture may change. Surrounding singularities (eigenvalues), there are now siginificant regions of the complex plane where the norm of the resolvent may still be very large. In effect, singularities appear smeared as a result of non normality and the very concept of eigenvalue becomes somewhat unappropriate. The concept of pseudo eigenvalue should be used instead. There are several variations on this theme but here we shall mostly be concerned with the ffl-pseudospectra of A defined by N. Trefethen  as
?ffl(A) = fz 2 C :k (zI ? A)?1 k2>= ffl?1g: (1)
1Thinking Machines Corporation, 245 First Street, Cambridge MA 02142, firstname.lastname@example.org