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

Andrew Hassell, Simon Marshall


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.

Schrödinger operator, negative eigenvalues, eigenvalue asymptotics.

Math Review Classification
35P20, 47A10

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28 pages

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