Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Trace identities for commutators, with applications to the distribution of eigenvalues

Authors: Evans M. Harrell II and Joachim Stubbe
Journal: Trans. Amer. Math. Soc. 363 (2011), 6385-6405
MSC (2010): Primary 81Q10, 35J25, 35P15, 35P20, 58C40
Published electronically: July 18, 2011
MathSciNet review: 2833559
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue $ \lambda_{N+1}$ in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrödinger operator on an immersed manifold of dimension $ d$, we derive a version of Reilly's inequality bounding the eigenvalue $ \lambda_{N+1}$ of the Laplace-Beltrami operator by a universal constant times $ \Vert h\Vert _{\infty}^2 N^{2/d}$.

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

  • 1. M. S. Ashbaugh, The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang, in Spectral and inverse spectral theory (Goa, 2000), Proc. Indian Acad. Sci. Math. Sci. 112 (2002) 3-30. MR 1894540 (2004c:35302)
  • 2. H. A. Bethe and R. W. Jackiw, Intermediate quantum mechanics, 2nd edn., W.A. Benjamin, New York, 1968. MR 0161579 (28:4783)
  • 3. M. Sh. Birman and M.Z. Solomyak, Estimates for the Number of Negative Eigenvalues of the Schrödinger Operator and Its Generalizations, in Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations, M.Sh. Birman, editor, Advances in Soviet Mathematics, Vol. 7, 1991, AMS, pp. 1-56. MR 1306507 (95h:35161)
  • 4. Ph. Blanchard and J. Stubbe, Bound states for Schrödinger Hamiltonians: Phase Space Methods and Applications. Rev. Math. Phys. 35, 504-547 (1996). MR 1405763 (97g:35117)
  • 5. Q. -M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005) 445-460. MR 2115463 (2005i:58038)
  • 6. A. El Soufi, E. M. Harrell, and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Amer. Math. Soc. 361 (2009) 2337-2350. MR 2471921
  • 7. A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du Laplacien et volume conforme, Math. Ann. 276 (1986) 257-267. MR 854009 (87j:53088)
  • 8. A. El Soufi, and S. Ilias, Une inégalité du type ``Reilly'' pour les sous-variétés de l'espace hyperbolique, Comment. Math. Helv. 67 (1992) 167-181. MR 1161279 (93i:53059)
  • 9. A. El Soufi and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature, Commun. Math. Phys. 208 (2000) 761-770. MR 1736334 (2001g:58050)
  • 10. E. M. Harrell, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Commun. in Partial Diff. Eqs. 32 (2007) 401-413. MR 2304154 (2008i:35041)
  • 11. E. M. Harrell and L. Hermi, Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues. J. Funct. Analysis 254 (2008) 3173-3191. MR 2418623 (2009f:47067)
  • 12. E. M. Harrell and L. Hermi, On Riesz Means of Eigenvalues. preprint 2007.
  • 13. E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349 (1997) 1797-1809. MR 1401772 (97i:35129)
  • 14. E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, preprint 2008, available as arXiv:0808.1133.
  • 15. E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, J. Funct. Anal. 256 (2009) 3977-3995. MR 2521917
  • 16. L. Hermi, Two new Weyl-type bounds for the Dirichlet Laplacian, Trans. Amer. Math. Soc. 360 (2008) 1539-1558. MR 2357704 (2009e:35035)
  • 17. D. Hundertmark, Some bound state problems in quantum mechanics. Proc. of Symposia in Pure Mathematics 76, part 1, Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday, Gesztesy et al., editors, 463-496, AMS, 2007. MR 2310215 (2008h:81047)
  • 18. A. A Ilyin, Lieb-Thirring inequalities on the $ N$-sphere and in the plane, and some applications. Proc. London Math. Soc. 67 (1963)159-182. MR 1218124 (94d:35129)
  • 19. A. A Ilyin, Best constants for a class of polymultiplicative inequalities for derivatives, Mat. Sbornik 189 (1998) 1295-1333. MR 1680856 (2000d:46042)
  • 20. A. A Ilyin, Lieb-Thirring integral inequalities and their applications to attractors of the Navier-Stokes equations, Mat. Sbornik 196 (2006) 29-61. MR 2141323 (2006d:35183)
  • 21. E. Krätzel, Lattice Points. Mathematics and its Applications (East European Series). Kluwer, Dordrecht, 1988. MR 998378 (90e:11144)
  • 22. P. Kuchment, Floquet theory for partial differential equations, Operator theory, advances and applications 60, Boston and Basel, Birkhäuser, 1993. MR 1232660 (94h:35002)
  • 23. A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997) 531-545. MR 1491551 (99a:35027)
  • 24. Laptev, A. and Weidl, T., Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184 (2000) 87-111. MR 1756570 (2001c:35173)
  • 25. M. Levitin and L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002) 425-445. MR 1923409 (2003g:47040)
  • 26. F. Odeh and J. B. Keller, Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals, J. Math. Phys. 5 (1964) 1499-1504. MR 0168924 (29:6180)
  • 27. M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators. Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • 28. R. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977) 525-533. MR 0482597 (58:2657)
  • 29. G. Rozenblum and M. Solomyak, Counting Schrödinger bound states: semiclassics and beyond, arXiv:0803.3138v2 [math.SP] 22 Nov 2008.
  • 30. Yu. Safarov, Lower bounds for the generalized counting function, in: The Maz'ya Anniversary Collection, vol. 2, Rostock, 1998, in: Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 275-293. MR 1747899 (2001d:47036)
  • 31. M. M. Skriganov, Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators, Proceedings of the Steklov Institute of Mathematics 171(1987). American Mathematical Society Providence. MR 798454 (87h:47110)
  • 32. J. Stubbe, Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities, J. Eur. Math. Soc. 12 (2010) 1347-1353. MR 2734344
  • 33. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Math. Sci. 58. New York: Springer-Verlag, 1997. MR 1441312 (98b:58056)
  • 34. S. Wang, Generalization of the Thomas-Reiche-Kuhn and the Bethe sum rules. Phys. Rev. A 60 (1999) 262-266.
  • 35. H. C. Yang, Estimates of the difference between consecutive eigenvalues, preprint 1995 (revision of International Centre for Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 81Q10, 35J25, 35P15, 35P20, 58C40

Retrieve articles in all journals with MSC (2010): 81Q10, 35J25, 35P15, 35P20, 58C40

Additional Information

Evans M. Harrell II
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0610

Joachim Stubbe
Affiliation: Department of Mathematics, Ecole Polytechnique Federale de Lausanne, IMB-FSB, Station 8, CH-1015 Lausanne, Switzerland

Received by editor(s): November 11, 2009
Published electronically: July 18, 2011
Article copyright: © Copyright 2011 by the authors

American Mathematical Society