  The discrete Schr\"odinger equation with the Dirichlet boundary condition is
considered on a half-line lattice when the potential is real valued and
compactly supported. The inverse problem of recovery of the potential from the
so-called transmission eigenvalues is analyzed. The Marchenko method and the
Gel'fand-Levitan method are used to solve the inverse problem uniquely, except
in one "unusual" case where the sum of the transmission eigenvalues is equal to
a certain integer related to the support of the potential. It is shown that in
the unusual case there may be a unique solution corresponding to certain sets
of transmission eigenvalues, there may be a finite number of distinct
potentials for some sets of transmission eigenvalues, or there may be
infinitely many potentials for some sets of transmission eigenvalues. The
theory presented is illustrated with several explicit examples.
