Eigenvalues of Schroedinger operators with potential asymptotically homogeneous of degree -2
Abstract
We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function
N
L
(E)
, the number of bound states of the operator
L=Δ+V
in $\R^d$ below
−E
. Here
V
is a bounded potential behaving asymptotically like
P(ω)
r
−2
where
P
is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator
Δ
S
d−1
+P
on the sphere has negative eigenvalues
−
μ
1
,...,−
μ
n
less than
−(d−2
)
2
/4
, we prove that
N
L
(E)
may be estimated as
N
L
(E))=
log(
E
−1
)
2π
∑
i=1
n
μ
i
−(d−2
)
2
/4
−
−
−
−
−
−
−
−
−
−
−
−
√
+O(1);
thus, in particular, if there are no such negative eigenvalues then
L
has a finite discrete spectrum. Moreover, under some additional assumptions including that
d=3
and that there is exactly one eigenvalue
−
μ
1
less than -1/4, with all others
>−1/4
, we show that the negative spectrum is asymptotic to a geometric progression with ratio $\exp(-2\pi/\sqrt{\mu_1 - \qtr})$.