On the Preservation of Absolutely Continuous Spectrum for Schrödinger Operators

We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered. In this paper, we consider the problem of preservation of a.c. spectrum for Schrödinger operators when the perturbation (potential) is decaying slowly at infinity. In a recent years, lots of results in this direction (including analysis of Jacobi matrices and Dirac operators) were obtained [10, 2, 9, 14, 20, 24, 3]. The proofs were of analytical flavor and had roots in the approximation theory, in particular, the theory of orthogonal polynomials of one variable [25, 13]. In the one-dimensional case, most of the questions were answered and we have a rather complete understanding of the picture. In the higher dimension, we know much less. In a series of papers [15, 16, 4, 5], authors were partially motivated by the following conjecture Conjecture. (B. Simon, [22]) Consider H = −∆+ V (x), x ∈ R with ∫ Rd V (x) |x|d−1 + 1 < +∞ Show that σac(H) = R . In spite of some progress in the field, this conjecture is still an open problem. In the current paper, we consider analogous problems, but on different domains. In the first section, we prove one general result on preservation of absolutely continuous spectrum for block matrices. We also apply this result to Schrödinger operator on the strip. The second section is devoted to the discrete Schrödinger operator on Caley tree (Bethe lattice) and the corresponding L conjecture. We Date: December 22, 2004.