Метод подобных операторов в исследовании спектральных свойств разностных операторов с растущим потенциалом

We study a difference operator with a general growing potential. We obtain asymptotic formulas for the eigenvalues, eigenvectors, and the spectral projections of such operators. We also obtain estimates for the equiconvergence of spectral decompositions. For an invertible difference operator, we estimate off-diagonal decay of the matrix of the inverse operator and the Fourier coefficients of the eigenvectors.