Monotonicity of the eigenvalues of the two-particle Schr\"odinger operatoron a lattice

J. I. Abdullaev ; A. M. Khalkhuzhaev ; L. S. Usmonov

Nanosistemy: fizika, himiâ, matematika, Tome 12 (2021) no. 6, pp. 657-663.

Résumé

We consider the two-particle Schrödinger operator $H(\mathbf{k})$, ($\mathbf{k}\in\mathbf{T^3}\equiv(-\pi,\pi]^3$) is the total quasimomentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice $\mathbf{Z}^3$. It is proved that the number $N(\mathbf{k})\equiv N(k^{(1)},k^{(2)},k^{(3)})$ of eigenvalues below the essential spectrum of the operator $H(\mathbf{k})$ is nondecreasing function in each $k^{(i)}\in[0,\pi]$, $i=1,2,3$. Under some additional conditions potential $\hat{v}$, the monotonicity of each eigenvalue $z_n(\mathbf{k})\equiv z_n(k^{(1)},k^{(2)},k^{(3)})$ of the operator $H(\mathbf{k})$ in $k^{(i)}\in[0,\pi]$ with other coordinates $\mathbf{k}$ being fixed is proved.

Détail

Keywords: two-particle Schrödinger operator, Birman–Schwinger principle, total quasimomentum, monotonicity of the eigenvalues.