Existence and analyticity of bound states of a~two-particle Schr\"odinger operator on a~lattice

S. N. Lakaev ; S. S. Ulashov

Teoretičeskaâ i matematičeskaâ fizika, Tome 170 (2012) no. 3, pp. 393-408

Résumé

We consider the two-particle discrete Schrödinger operator $H_\mu(K)$ corresponding to a system of two arbitrary particles on a $d$-dimensional lattice $\mathbb Z^d$, $d\ge3$, interacting via a pair contact repulsive potential with a coupling constant $\mu>0$ ($K\in\mathbb T^d$ is the quasimomentum of two particles). We find that the upper (right) edge of the essential spectrum can be either a virtual level (for $d=3,4)$ or an eigenvalue (for $d\ge5)$ of $H_\mu(K)$. We show that there exists a unique eigenvalue located to the right of the essential spectrum, depending on the coupling constant $\mu$ and the two-particle quasimomentum $K$. We prove the analyticity of the corresponding eigenstate and the analyticity of the eigenvalue and the eigenstate as functions of the quasimomentum $K\in\mathbb T^d$ in the domain of their existence.

Détail

Keywords: discrete Schrödinger operator, two-particle system, Hamiltonian, contact repulsive potential, virtual level, eigenvalue, lattice.