Ramiza Kh. Djumanova | Searcharxiv

arXiv2006

Low energy effects for a family of Friedrichs models

A family of Friedrichs models with rank one perturbations $h_μ(p),$ $p \in (-π,π]^3,μ>0$ associated to a system of two particles on the lattice $\Z^3$ is considered. The existence of a unique strictly positive eigenvalue below the bottom of the essential spectrum of $h_μ(p)$ for all nontrivial values $p \in (-π,π]^3$ under the assumption that $h_μ(0)$ has either a zero energy resonance (virtual level) or a threshold eigenvalue is proved. Low energy asymptotic expansion for the Fredholm determinant associated to family of Friedrichs models is obtained.

math.SP↗

arXiv2005

On the essential and discrete spectrum of a model operator related to three-particle discrete Schrödinger operators

A model operator $H$ corresponding to a three-particle discrete Schrödinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_α(p),$ $α=1,2,$ $p \in \T^3=(-π,π]^3.$ The following results are proven: 1) The operator $H$ has a finite number of eigenvalues lying below the bottom of the essential spectrum in any of the following cases: (i) both operators $h_α(0), α=1,2,$ have a zero eigenvalue; (ii) either $h_1(0)$ or $h_2(0)$ has a zero eigenvalue. 2) The operator $H$ has infinitely many eigenvalues lying below the bottom and accumulating at the bottom of the essential spectrum, if both operators $h_α(0),α=1,2,$ have a zero energy resonance.

math-ph↗