We present the basics of two-body quantum-mechanical scattering theory and the theory of quantum resonances. The wave operators and S-matrix are constructed for smooth, compactly-supported potential perturbations of the Laplacian. The meromorphic continuation of the cut-off resolvent is proved for the same family of Schrödinger operators. Quantum resonances are defined as the poles of the merom...

We prove upper bounds on the order and degree of the polynomials involved in a resolvent representation of the prime differential ideal associated with a polynomial differential system for a particular class of ordinary first order algebraic-differential equations arising in control theory. We also exhibit a probabilistic algorithm which computes this resolvent representation within time polyno...

In this note, we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization, called the generalised companion matrix. A special resolvent decomposition of the regular polynomial matrix is proposed which is based on the Weierstrass canonical form of this generalised companion matrix and the solution of a ...

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdös-Renyi graphs and preferential attachment graphs. We sketch examples of a...

We prove a variant of Krein’s resolvent formula for self-adjoint extensions given by arbitrary boundary conditions. A parametrization of all such extensions is suggested with the help of two bounded operators instead of multivalued operators and selfadjoint linear relations.

It is shown that the right shift semigroup on L(R+) does not satisfy the weighted Weiss conjecture for α ∈ (0, 1). In other words, αadmissibility of scalar valued observation operators cannot always be characterised by a simple resolvent growth condition. This result is in contrast to the unweighted case, where 0-admissibility can be characterised by a simple growth bound. The result is proved ...