Bound states of the discrete Schrodinger equation with compactly supported potentials

Abstract

The discrete Schrodinger operator is considered on the half-line lattice $n\in \{1,2,3,\dots\}$ with the Dirichlet boundary condition at $n=0$. 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 inequalities $0\le N\le b$. It is shown that for each fixed nonnegative integer k in the set $\{0,1,2,\ldots,b\}$, there exist infinitely many potentials in class $\mathcal{A}_b$ for which the corresponding Schrodinger 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 Schrodinger operator. The theory presented is illustrated with some explicit examples.

Keywords

• Discrete Schrodinger operator
• half-line lattice
• bound states
• resonances
• compactly-supported potential
• number of bound states