Let
L
0
=
−
Δ
+
V
(
x
1
)
,
L
=
L
0
+
V
p
(
x
)
{L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)
be selfadjoint in
L
2
(
R
n
)
{L^2}({R^n})
. Here
V
,
V
p
V,{V_p}
are real functions,
V
(
x
1
)
V({x_1})
depends only on the first coordinate. Existence of the wave-operators
W
±
(
L
,
L
0
)
=
s
-
lim
t
→
±
∞
exp
(
i
t
L
)
exp
(
−
i
t
L
0
)
{W_ \pm }\,(L,{L_0}) = s \text {-} {\lim _{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0})
is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of
−
d
2
/
d
t
2
+
V
(
t
)
-{d^2}/d{t^2} + V(t)
in
L
2
(
R
)
{L^2}(R)
. Its absolute continuity is proved as well as a suitable eigenfunction expansion.
V
V
is a "Stark-like" potential. In particular, the cases
V
(
x
1
)
=
(
−
sgn
x
1
)
|
x
1
|
α
,
0
>
α
⩽
2
V({x_1}) = ( - \operatorname {sgn}{x_1})|{x_1}\,{|^\alpha },0 > \alpha \leqslant 2
, are included.
V
p
{V_p}
may be taken as the sum of an
L
2
{L^2}
-function and a function satisfying growth conditions in the
+
x
1
+ {x_1}
direction.
V
p
(
x
)
=
|
x
|
−
1
{V_p}(x) = |x|^{ - 1}
is included.