On the infinite number of eigenvalues of the two-particle Schrödinger operator on a lattice

J. I. Abdullaev ; A. M. Khalkhuzhaev ; Yu. S. Shotemirov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2024), pp. 3-11

Résumé

We consider the Schrödinger operator $H(\mathbf{k})=H_0(\mathbf{k})-V, \mathbf{k}\in \mathbb{T}^2,$ associated with a system of two particles on a two-dimensional lattice. It is shown that the subspaces of even as well as odd functions are invariant under operator $H(\mathbf{k}).$ The sets of quasimomenta $\mathcal{K}(1),$ $\mathcal{K}(2)$ and the class of potentials $\mathrm{P}(1),$ $\mathrm{P}(2)$ are described, for which the operator $H(\mathbf{k})$ has infinite number of eigenvalues $z_n(\mathbf{k}), n\in \mathbb{Z}_+$, for $\mathbf{k}\in \mathcal{K}(j), \hat{v}\in \mathrm{P}(j)$. The explicit form of $z_n(\mathbf{k})$ and the rate of convergence of the sequence $z_n(\mathbf{k})$ to the bottom of the essential spectrum are found.

Détail

Keywords: lattice, Hamiltonian, Schrödinger operator, quasimomentum, width of the continuous spectrum, potential, eigenvalue, eigenfunction.

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