The threshold effects for a family of Friedrichs models under rank one perturbations

Abstract

A family of Friedrichs models under rank one perturbations $h_{\mu }(p),$ $p\in (-\pi ,\pi {]}^3$, $\mu >0,$ associated to a system of two particles on the three dimensional lattice ${\mathbb{Z}}^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_{\mu }(p)$ for all nontrivial values of $p$ under the assumption that $h_{\mu }(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.

15 pages

References in corpus (1)

• On the number of eigenvalues of a model operator associated to a system of three-particles on lattices