Schrödinger operator with decreasing potential in a cylinder

The Schrödinger operator − Δ + V ( x , y ) -\Delta + V(x,y) is considered in a cylinder R m × U \mathbb {R}^m \times U , where U U is a bounded domain in R d \mathbb {R}^d . The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, | V ( x , y ) | ≤ C ⟨ x ⟩ − ρ |V(x,y)| \le C \langle x\rangle ^{-\rho } . If ρ > 1 \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.