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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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 $\exp (-2\pi /\sqrt {\mu _1 - \frac {1}{4}})$.
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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