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.