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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}})$.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, USA, 1964.
  • 2. T. Christiansen and M. Zworski, Spectral asymptotics for manifolds with cylindrical ends, Ann. Inst. Fourier (Grenoble) 45 (1995), 251-263. MR 1324132 (96d:35100)
  • 3. Y. Colin de Verdière, Pseudo-laplaciens. I, II. Ann. Inst. Fourier (Grenoble) 32 (1982), 275-286; 33 (1983), 87-113. MR 688031 (84k:58221)
  • 4. C. Fefferman, D. H. Phong, The uncertainty principle and sharp Garding inequalities, Comm. Pure Appl. Math. 34 (1981), 285-331. MR 611747 (82j:35140)
  • 5. C. Fefferman, D. H. Phong, On the asymptotic eigenvalue distribution of a pseudodifferential operator, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 10, 5622-5625. MR 589278 (82h:35101)
  • 6. R. Froese, I. Herbst, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, $ L^{2}$-lower bounds to solutions of one-body Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 1-2, 25-38. MR 723095 (86a:35044)
  • 7. A. Grigor'yan, S.-T. Yau, Isoperimetric properties of higher eigenvalues of elliptic operators, Amer. J. Math. 125 (2003), no. 4, 893-940. MR 1993744 (2004g:58041)
  • 8. L. Hörmander, The analysis of linear partial differential operators, vol. III, Springer-Verlag, Berlin, 1985.
  • 9. T. Kato, Some results on potential scattering, Proc. Intern. Conf. on Funct. Anal. and related topics, 1969, University of Tokyo Press, Tokyo, 206-215. MR 0268713 (42:3610)
  • 10. W. Kirsch and B. Simon, Corrections to the Classical Behaviour of the Number of Bound States of Schrödinger Operators, Annals of Physics, 183 (1988), 122-130. MR 952875 (90b:35065)
  • 11. P. Lax and R. Phillips, Scattering theory for automorphic functions, Ann. of Math. Stud., 87, Princeton Univ. Press, Princeton, N.J., 1976. MR 0562288 (58:27768)
  • 12. R. Melrose, The Atiyah-Patodi-Singer index theorem, Wellesley, Mass.: A.K. Peters, 1993. MR 1348401 (96g:58180)
  • 13. M. Reed and B. Simon, Methods of modern mathematical physics, vol. 4, Academic Press, San Diego, 1979.
  • 14. G. Rozenblum, M.Solomyak, On the number of negative eigenvalues for the two-dimensional magnetic Schrödinger operator, in Differential operators and spectral theory, 205-217, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. MR 1730514 (2000j:35214)
  • 15. D. Yafaev, Scattering theory: some old and new problems, Springer Lecture Notes in Mathematics 1735, Springer, Berlin, 2000. MR 1774673 (2001j:81248)

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Additional Information

Andrew Hassell
Affiliation: Department of Mathematics, The Australian National University, ACT 0200, Australia

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

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.

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