The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice

We consider the Hamiltonian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat h_{\mu \lambda } ,\mu ,\lambda \geqslant 0 $$\end{document}, describing the motion of one quantum particle on a three-dimensional lattice in an external field. We investigate the number of eigenvalues and their arrangement depending on the value of the interaction energy for µ ≥ 0 and λ ≥ 0.