Imbedded singular continuous spectrum for Schrödinger operators

Abstract

We construct examples of potentials V ( x ) V(x) satisfying | V ( x ) | ≤ h ( x ) 1 + x , |V(x)| \leq \frac {h(x)}{1+x}, where the function h ( x ) h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if | V ( x ) | ≤ B 1 + x , |V(x)| \leq \frac {B}{1+x}, the singular continuous spectrum is empty. Therefore our result is sharp.