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The discrete Schr\"odinger equation with the Dirichlet boundary condition isconsidered on a half-line lattice when the potential is real valued andcompactly supported. The inverse problem of recovery of the potential from theso-called transmission eigenvalues is analyzed. The Marchenko method and theGel'fand-Levitan method are used to solve the inverse problem uniquely, exceptin one "unusual" case where the sum of the transmission eigenvalues is equal toa certain integer related to the support of the potential. It is shown that inthe unusual case there may be a unique solution corresponding to certain setsof transmission eigenvalues, there may be a finite number of distinctpotentials for some sets of transmission eigenvalues, or there may beinfinitely many potentials for some sets of transmission eigenvalues. Thetheory presented is illustrated with several explicit examples.

Inverse problem with transmission eigenvalues for the discrete Schr\"odinger equation