| Summary: | 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.
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