The similar operator method is used for the spectral analysis of the closed difference operator of the form (A x)(n) = x(n + 1) + x(n − 1) − 2x(n) + a(n)x(n), n ∈ Z under consideration in the Hilbert space l2(Z) of bilateral sequences of complex numbers, with a growing potential a : Z → C. The asymptotic estimates of eigenvalue, eigenvectors, spectral estimation of equiconvergence applications for the test operator and the operator of multiplication by a sequence a : Z → C.
