Asymptotics of the eigenvalues of a~discrete Schr\"odinger operator with zero-range potential

S. N. Lakaev ; Sh. Yu. Kholmatov

Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 946-966

Résumé

We consider a family of discrete Schrödinger operators $H_{\mu}(k)$, $k\in\mathfrak{G}\subset\mathbb{T}^d$. These operators are associated with the Hamiltonian ${H}_{\mu}$ of a system of two identical quantum particles (bosons) moving on the $d$-dimensional lattice $\mathbb{Z}^d$, $d\geqslant 3$, and interacting by means of a pairwise zero-range (contact) attractive potential $\mu>0$. It is proved that for any $k\in\mathfrak{G}$ there is a number $\mu(k)>0$ which is a threshold value of the coupling constant; for $\mu>\mu(k)$ the operator $H_{\mu}(k)$, $k\in\mathfrak{G}\subset\mathbb{T}^d$, has a unique eigenvalue $z(\mu, k)$ placed to the left of the essential spectrum. The asymptotic behaviour of $z(\mu, k)$ is found as $\mu\to\mu(k)$ and as $\mu\to+\infty$ and also as $k\to k^*$ for every value of the quasi-momentum $k^*=k^*(\mu)$ belonging to the manifold $\{k\in\mathfrak{G}\colon\mu(k)=\mu\}$, where $\mu\in\bigl(\inf_{k\in\mathfrak{G}}\mu(k),\sup_{k\in\mathfrak{G}}\mu(k)\bigr)$.

Détail

Keywords: discrete Schrödinger operator, Hamiltonian system of two particles, zero-range (contact) potential, eigenvalue, asymptotic behaviour.