Abstract. We give an overview for spectral analysis of non-self adjoint operators from a mathematical standpoints. Among other things, we concentrate our considerations on operators that appeared for Schrödinger and wave equations. To these examples, we describe some results of the relation between spectral structure of generator and asymptotic behavior of solutions (energy decay and scattering...

We consider quite general h-pseudodifferential operators on R with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different. Résumé Nous considérons des opérateurs h-pseudodifférentiels as...

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operato...

We recall the form factors f (j) N,N corresponding to the λ-extension C(N,N ;λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a “Russian-doll” nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differen...

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients s...

We give necessary and sufficient conditions for local solvability and hypoellipticity of some classes of infinitely degenerate elliptic differential operators.

We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint n× n matrices.