On the spectrum of lattice Schrödinger operators with deterministic potential. II.
The results of this paper extend those of the author in [ibid. 87, 37--75 (2002; Zbl 1033.47024)]. For one-dimensional lattice Schrödinger operators on \(L^2({\mathbb Z})\) with two frequencies, \[ H_{\omega}(x_1, x_2) = v(x_1 + n\omega_1, x_2 + n\omega_2)\delta_{n,n'} + \Delta, \] \(v\) being a trigonometric polynomial on \({\mathbb T}^2\), and \(\Delta\) the lattice Laplacian, the presence of a point spectrum, a continuous spectrum, and a gap is investigated: if \(v\) has a non-degenerate maximum and \(\omega\) belongs to a certain set of positive measure, it is shown that \(H_{\omega}(0)\) has a point spectrum whose closure has positive measure. For the special case \(v = \lambda\cos(x_1 + n\omega_1) + \tau\cos(x_2 + n\omega_2)\) with \(0<\lambda<2\), small \(\omega_i\), and a very small \(\tau > 0\), there is also an absolutely continuous spectrum. The existence of spectral gaps is proved for \(H_{\omega}(0)\) when \(\lambda = \tau > 0\) and small. The methods used in this paper are similar to those applied in the former one.
one-dimensional lattice Schrödinger operators
scientific article; zbMATH DE number 1998321