On a conjecture by Y. Last

Conjecture 0.1. Do the following conditions: vn → 0 and vn+q − vn ∈ l (q ∈ Z– fixed) guarantee that σac(J) = [−2, 2]? The symbol σac(J) conventionally denotes the absolutely continuous (a.c.) spectrum of self-adjoint operator J . In this paper, we give an affirmative answer to this question. The manuscript consists of two sections. The first one is mostly algebraic, it contains the determinantal formula for the so-called transmission coefficient that allows us to immediately treat the case q = 1. In the second part, we show how asymptotical methods for difference equations provide the solution for any q. The appendix contains an elementary lemmas for harmonic functions which are used in the paper. Recently, many results on the characterization of parameters in the Jacobi matrix through the spectral data were obtained and the l–condition on coefficients was often involved in one form or another (see, e.g., [1, 4, 6, 10]). This paper makes the next step in this direction by developing the technique suggested in [2]. We will use notations: (δv)n = vn+1 − vn, (δv)n = vn+q − vn, χj∈M is the characteristic function of the set M . For the sequence α ∈ l, the symbol ‖α‖p denotes its norm in l. As usual, the symbol C denotes the positive constant which might take different values in different formulas. For any matrix B ∈ C, the symbol ‖B‖ will denote its operator norm in C. Consider a linear bounded operator A acting in the Hilbert space. Assume that it is Hilbert-Schmidt, i.e. A ∈ S2. Then, we define the regularized determinant by the formula (see, e.g., [9]) det 2(I +A) = det(I +R2(A))