Existence condition of an eigenvalue of the three particle Schr\"{o}dinger operator on a lattice

Zh. I. Abdullaev ; A. M. Khalkhuzhaev ; I. A. Khujamiyorov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2023), pp. 3-25

Résumé

We consider the three-particle discrete Schrödinger operator $H_{\mu,\gamma}(\mathbf{K}),$ $\mathbf{K}\in\mathbb{T}^3$ associated to a system of three particles (two particle are fermions with mass $1$ and third one is an another particle with mass $m=1/\gamma1$ ) interacting through zero range pairwise potential $\mu>0$ on the three-dimensional lattice $\mathbb{Z}^3.$ It is proved that for $\gamma \in (1,\gamma_0)$ ($\gamma_0\approx 4,7655$) the operator $H_{\mu,\gamma}(\boldsymbol{\pi}),$ $\boldsymbol{\pi}=(\pi,\pi,\pi),$ has no eigenvalue and has only unique eigenvalue with multiplicity three for $\gamma>\gamma_0$ lying right of the essential spectrum for sufficiently large $\mu.$

Détail

Keywords: Schrödinger operator on a lattice, Hamiltonian, zero-range, eigenvalue, quasimomentum, invariant subspace, Faddeev operator.
Mots-clés : fermion