The spectrum of differential operators with almost constant coefficients II

Abstract

The absolutely continuous spectrum of differential operators of the form Ly = w(-1)Sigma(k=0)(n)(-1)(k)(p(k)y((k)))((k)) on L-2([9, infinity), w) is determined. With p(n)(x), w(x) > 0 the coefficients p(k) are assumed to satisfy (p) over tilde (k) (x) = (p(k) y(2k)w(-1))(x) --> c(k), y = (w (.) p(n)(-1))(1/2n). If the coefficients satisfy some additional smoothness and decay conditions, the absolutely continuous part H-ac of any self-adjoint extension of L is unitarily equivalent to the operator of multiplication by P(x) = Sigma(0)(n) c(k)x(2k) on L-2([0, infinity)). Several extensions of this result as well as examples are shown. (C) 2002 Elsevier Science B.V. All rights reserved.