Schrödinger operator with decreasing potential in a cylinder

Abstract:

The Schrödinger operator $-\Delta + V(x,y)$ is considered in a cylinder $\mathbb {R}^m \times U$, where $U$ is a bounded domain in $\mathbb {R}^d$. The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, $|V(x,y)| \le C \langle x\rangle ^{-\rho }$. If $\rho > 1$, then the wave operators exist and are complete; the Birman invariance principle and the limiting absorption principle hold true; the absolute continuous spectrum fills the semiaxis; the singular continuous spectrum is empty; the eigenvalues can accumulate to the thresholds only.