The eigenvalue problem for the resonances of the infinite-dimensional Friedrichs model on the positive half line with Hilbert-Schmidt perturbations

TL;DR

This paper constructs a Gelfand triplet for the Friedrichs model's Hamiltonian on the positive half line, explicitly calculates eigenantilinear forms, and links resonances to decay semigroups and Gamov vectors, revealing detailed spectral properties.

Contribution

It introduces a Gelfand triplet framework for the Friedrichs model with Hilbert-Schmidt perturbations, explicitly computes eigenantilinear forms, and connects resonances to decay semigroups and Gamov vectors.

Findings

Resonances correspond to eigenvalues of an extended Hamiltonian.

Eigenantilinear forms are explicitly calculated and linked to Gamov vectors.

The scattering matrix has simple poles with Laurent parts expressed via Gamov vectors.

Abstract

A Gelfand triplet for the Hamiltonian H of the infinite-dimensional Friedrichs model on the positive half line with Hilbert-Schmidt perturbations is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are eigenvalues of the extension H^{\times} of H. The corresponding eigenantilinear forms are calculated explicitly. Using the wave matrices for the Abelian wave (M\"oller) operators the corresponding eigenantilinear forms for the unperturbed Hamiltonian $H_{0}$ turn out to be of pure Dirac type and can be characterized by their corresponding Gamov vector which is uniquely determined by restriction to the intersection of the Gelfand space for $H_{0}$ with $P_{+}H^{2}_{+}$, where $H^{2}_{+}$ is the Hardy space of the upper half plane. Simultaneously, this restriction yields a truncation of the unitary evolution $t\to e^{-itH_{0}}$ to the well-known decay semigroup for $t\geq 0$ of the Toeplitz type on $P_{+}H^{2}_{+}$. That is, exactly those eigenvectors $\lambda\to k(\lambda-\zeta)^{-1}$, $k$ element of the multiplicity space K, of the decay semigroup have an extension to an eigenantilinear form for $H_{0}$ hence for H if $\zeta$ is a resonance and k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinear form. Moreover, the scattering matrix which is meromorphic in the lower half plane has only simple poles there and the main part of its Laurent representation is a linear combination of Gamov vectors.