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The discrete Schr\"odinger operator with the Dirichlet boundary condition isconsidered on the half-line lattice $n\in \{1,2,3,\dots\}.$ It is assumed thatthe potential belongs to class $\mathcal A_b,$ i.e. it is real valued, vanisheswhen $n>b$ 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 fixednonnegative integer $k$ in the set $\{0,1,2,\dots,b\},$ there exist infinitelymany potentials in class $\mathcal A_b$ for which the correspondingSchr\"odinger operator has exactly $k$ bound states. Some auxiliary results arepresented to relate the number of bound states to the number of real resonancesassociated with the corresponding Schr\"odinger operator. The theory presentedis illustrated with some explicit examples.

On the bound states of the discrete Schr\"odinger equation with compactly supported potentials