Inverse and Direct Spectral Problems for a Class of Lattice Hamiltonians

We consider the solvability of the inverse and direct spectral problems for a class of limit-periodic operators on a lattice $\mathbb{Z}^d$ for any $d\ge 1$, generalizing the lattice Schrödinger operator $H = \Delta + U$ in $\ell^2(\mathbb{Z}^d)$ with an external potential $U$. This problem was intensively studied in the 1980s where the solutions were shown to exist under the assumption of subexponential decay rate of the so-called small denominators. Since then, this assumption has appeared in a number of mathematical works. We show that it can be relaxed to a weaker assumption of exponential decay of small denominators for any arbitrarily large positive decay exponent. As in many prior works, we prove that the admissible limit-periodic operators have an eigenbasis formed by exponentially decaying lattice functions, thus featuring the uniform exponential Anderson localization. To this end, we improve the existing techniques and, to a certain extent, simplify them, rendering the proofs more transparent. Some classes of operators are discussed in an explicit manner.