# Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree -2

## Andrew Hassell, Simon Marshall

__Abstract__

We strengthen and generalise a result of Kirsch and Simon on the
behaviour of the number of bound states of a Schrödinger operator L with
bounded potential behaving asymptotically like P(\omega)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. We give an O(1)
asymptotic formula for the counting function in terms of the negative
eigenvalues of the operator \Delta_{S^(n-1)}+P on the sphere. In particular,
if the spherical operator has no eigenvalues less than -(d-2)^2/4 then L has a
finite discrete spectrum. Moreover, under some additional assumptions including
that the spatial dimension is 3 and that there is exactly one eigenvalue less
than -1/4, with all others strictly greater than -1/4, we show that the
negative spectrum is asymptotic to a geometric progression.

__Keywords__

Schrödinger operator, negative eigenvalues, eigenvalue asymptotics.

__Math Review Classification__

35P20, 47A10

__Last Updated__

31/10/05

__Length__

28 pages

__Availability__

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