Generalized Eigenvectors for Resonances in the Friedrichs Model and Their Associated Gamov Vectors

TL;DR

This paper constructs a Gelfand triplet for the Friedrichs model Hamiltonian to explicitly identify resonances as generalized eigenvalues, linking them to Gamov vectors and decay semigroups.

Contribution

It introduces a new Gelfand triplet framework that precisely characterizes resonances and Gamov vectors for the Friedrichs model, including explicit calculations of eigen-antilinearforms.

Findings

Resonances are identified as eigenvalues of the Hamiltonian within the Gelfand triplet.

Gamov vectors are characterized by their restriction to Hardy space, linking to decay semigroups.

Explicit formulas for eigen-antilinearforms are derived, connecting resonances to decay dynamics.

Abstract

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on R with finite-dimensional multiplicity space K, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen-antilinearforms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen-antilinearforms on the Schwartz space S for the unperturbed Hamiltonian are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector, which is uniquely determined by restriction of S to the intersection of S with the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup of the Toeplitz type for the positive half line on the Hardy space. That is: exactly those pre-Gamov vectors (eigenvectors of the decay semigroup) have an extension to a generalized eigenvector of H if the eigenvalue is a resonance and if the multiplicity parameter k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinearform.