Asymptotic behavior of eigenvalues of the~two-particle discrete Schr\"odinger operator

J. I. Abdullaev ; B. U. Mamirov

Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 3, pp. 417-428

Résumé

We consider two-particle Schrödinger operator $H(k)$ on a three-dimensional lattice $\mathbb Z^3$ (here $k$ is the total quasimomentum of a two-particle system, $k\in\mathbb{T}^3:=(-\pi,\pi]^3$). We show that for any $k\in S=\mathbb{T}^3\setminus(-\pi,\pi)^3$, there is a potential $\hat v$ such that the two-particle operator $H(k)$ has infinitely many eigenvalues $z_n(k)$ accumulating near the left boundary $m(k)$ of the continuous spectrum. We describe classes of potentials $W(j)$ and $W(ij)$ and manifolds $S(j)\subset S$, $i,j\in\{1,2,3\}$, such that if $k\in S(3)$, $(k_2,k_3)\in(-\pi,\pi)^2$, and $\hat v\in W(3)$, then the operator $H(k)$ has infinitely many eigenvalues $z_n(k)$ with an asymptotic exponential form as $n\to\infty$ and if $k\in S(i)\cap S(j)$ and $\hat v\in W(ij)$, then the eigenvalues $z_{nm}(k)$ of $H(k)$ can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.

Détail

Keywords: Hamiltonian, total quasimomentum, Schrödinger operator, asymptotic behavior, eigenvalue, eigenfunction.