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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Andrew Hassell and Simon Marshall
Journal: Trans. Amer. Math. Soc. 360 (2008), 4145-4167
MSC (2000): Primary 35P20
Published electronically: March 13, 2008
MathSciNet review: 2395167
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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 = \Delta+V$ in $ \mathbb{R}^d$ below $ -E$. Here $ V$ is a 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. If the operator $ \Delta_{S^{d-1}}+P$ on the sphere $ S^{d-1}$ has negative eigenvalues $ -\mu_1,\ldots,-\mu_n$ less than $ -(d-2)^2/4$, we prove that $ N_L(E)$ may be estimated as

$\displaystyle N_L(E) = \frac{\log(E^{-1})}{2\pi}\sum_{i=1}^n \sqrt{\mu_i-(d-2)^2/4} +O(1).$

Thus, in particular, if there are no such negative eigenvalues, then $ L$ has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that $ d=3$ and that there is exactly one eigenvalue $ -\mu_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 - \frac{1}{4}})$.


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Additional Information

Andrew Hassell
Affiliation: Department of Mathematics, The Australian National University, ACT 0200, Australia
Email: hassell@maths.anu.edu.au

Simon Marshall
Affiliation: Department of Mathematics, The University of Auckland, Auckland 1142, New Zea-land
Address at time of publication: Department of Mathematics, Fine Hall, Princeton University, Washington Rd., Princeton, New Jersey 08544
Email: slm@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04479-6
PII: S 0002-9947(08)04479-6
Received by editor(s): June 8, 2006
Published electronically: March 13, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.