The KAM Approach to the Spectral Analysis of Markov Generators and Nonlocal Hamiltonians

We study inverse and direct spectral problems for a class of operators H = D + K acting in ℓ^2(ℤ^d), d ≥ 1. Here D is a multiplication operator, and K generalizes a lattice Laplacian and is given by a matrix (K_xy)_(x,y∈ℤ^d) with K_xx ≡ 0 and |K_x,y| ≤ ε e^(-c|x-y|^γ), where γ ∈ (0; 1), c > 0, 0 < ε ≪ 1. We streamline the existing KAM (Kolmogorov-Arnold-Moser) techniques and extend prior results, obtained for a simpler case γ = 1, to the case of a subexponential decay of the off-diagonal entries K_xy.