On the Singular Spectrum of Schrödinger Operators with Decaying Potential

The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the socalled Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal. Introduction We consider a Schrödinger operator Lqy = −y′′ + qy on the positive half-line R+ with boundary condition y′(0) + hy(0) = 0. Assume that q ∈ L(R+) is a real-valued function and h ∈ R∪{∞}. Denote the spectral measure of the operator Lq by ρ. Recently, Remling [22] proved the following theorem. Theorem 0.1 ([22], [23]). If |q(x)| ≤ C(1 + x)−β with 1/2 < β ≤ 1, then the support of the (possible) singular part of ρ has Hausdorff dimension less than or equal to 2(1− β). Actually, the stronger result was obtained, that is, the set of all positive spectral parameters such that the transfer matrix is not bounded at infinity has Hausdorff dimension less than or equal to 2(1− β). A result of the same nature was proved in [3]. We give a slightly weaker version here. Theorem 0.2 ([3]). Suppose that 0 < γ ≤ 1 and ∫ ∞ 0 (1 + s)q(s) ds < ∞. Then the support of the (possible) singular part of ρ has Hausdorff dimension less than or equal to 1− γ. This theorem readily implies the following statement. Received by the editors February 27, 2002 and, in revised form, November 4, 2003. 2000 Mathematics Subject Classification. Primary 34L05.