On the bound states of the discrete Schrödinger equation with compactly supported potentials

Abstract

The discrete Schrödinger operator with the Dirichlet boundary condition is considered on the half-line lattice $n\in \{1,2,3,\dots \}.$ It is assumed that the potential belongs to class ${\mathcal{A}}_b,$ i.e. it is real valued, vanishes when $n>b$ with $b$ being a fixed positive integer, and is nonzero at $n=b.$ The proof is provided to show that the corresponding number of bound states, $N,$ must satisfy the inequality $0\le N\le b.$ It is shown that for each fixed nonnegative integer $k$ in the set $\{0,1,2,\dots ,b\},$ there exist infinitely many potentials in class ${\mathcal{A}}_b$ for which the corresponding Schrödinger operator has exactly $k$ bound states. Some auxiliary results are presented to relate the number of bound states to the number of real resonances associated with the corresponding Schrödinger operator. The theory presented is illustrated with some explicit examples.

27 pages