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

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- by Andrew Hassell and Simon Marshall PDF
- Trans. Amer. Math. Soc.
**360**(2008), 4145-4167 Request permission

## 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 \[ N_L(E) = \frac {\log (E^{-1})}{2\pi }\sum _{i=1}^n \sqrt {\mu _i-(d-2)^2/4} +O(1).\] 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 $\exp (-2\pi /\sqrt {\mu _1 - \frac {1}{4}})$.## References

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## Additional Information

**Andrew Hassell**- Affiliation: Department of Mathematics, The Australian National University, ACT 0200, Australia
- MR Author ID: 332964
- 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
- Received by editor(s): June 8, 2006
- Published electronically: March 13, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**360**(2008), 4145-4167 - MSC (2000): Primary 35P20
- DOI: https://doi.org/10.1090/S0002-9947-08-04479-6
- MathSciNet review: 2395167