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

Harrell, Evans, II; Stubbe, Joachim

doi:10.1090/S0002-9947-2011-05252-9

Trace identities for commutators, with applications to the distribution of eigenvalues
HTML articles powered by AMS MathViewer

by Evans M. Harrell II and Joachim Stubbe PDF

Trans. Amer. Math. Soc. 363 (2011), 6385-6405

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 $\|h\|_{\infty }^2 N^{2/d}$.

References

Mark S. Ashbaugh, The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang, Proc. Indian Acad. Sci. Math. Sci. 112 (2002), no. 1, 3–30. Spectral and inverse spectral theory (Goa, 2000). MR 1894540, DOI 10.1007/BF02829638
Hans A. Bethe, Intermediate quantum mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1964. Notes by R. W. Jackiw. MR 0161579
M. Sh. Birman and M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1–55. MR 1306507
Ph. Blanchard and J. Stubbe, Bound states for Schrödinger Hamiltonians: phase space methods and applications, Rev. Math. Phys. 8 (1996), no. 4, 503–547. MR 1405763, DOI 10.1142/S0129055X96000172
Qing-Ming Cheng and Hongcang Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), no. 2, 445–460. MR 2115463, DOI 10.1007/s00208-004-0589-z
Ahmad El Soufi, Evans M. Harrell II, and Saïd Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2337–2350. MR 2471921, DOI 10.1090/S0002-9947-08-04780-6
A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du laplacien et volume conforme, Math. Ann. 275 (1986), no. 2, 257–267 (French). MR 854009, DOI 10.1007/BF01458460
A. El Soufi and S. Ilias, Une inégalité du type “Reilly” pour les sous-variétés de l’espace hyperbolique, Comment. Math. Helv. 67 (1992), no. 2, 167–181 (French). MR 1161279, DOI 10.1007/BF02566494
Ahmad El Soufi and Saïd Ilias, Second eigenvalue of Schrödinger operators and mean curvature, Comm. Math. Phys. 208 (2000), no. 3, 761–770. MR 1736334, DOI 10.1007/s002200050009
Evans M. Harrell II, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Comm. Partial Differential Equations 32 (2007), no. 1-3, 401–413. MR 2304154, DOI 10.1080/03605300500532889
Evans M. Harrell II and Lotfi Hermi, Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Anal. 254 (2008), no. 12, 3173–3191. MR 2418623, DOI 10.1016/j.jfa.2008.02.016
E. M. Harrell and L. Hermi, On Riesz Means of Eigenvalues. preprint 2007.
Evans M. Harrell II and Joachim Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1797–1809. MR 1401772, DOI 10.1090/S0002-9947-97-01846-1
E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, preprint 2008, available as arXiv:0808.1133.
Evans M. Harrell II and Selma Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Anal. 256 (2009), no. 12, 3977–3995. MR 2521917, DOI 10.1016/j.jfa.2008.12.008
Lotfi Hermi, Two new Weyl-type bounds for the Dirichlet Laplacian, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1539–1558. MR 2357704, DOI 10.1090/S0002-9947-07-04254-7
Dirk Hundertmark, Some bound state problems in quantum mechanics, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 463–496. MR 2310215, DOI 10.1090/pspum/076.1/2310215
A. A. Ilyin, Lieb-Thirring inequalities on the $N$-sphere and in the plane, and some applications, Proc. London Math. Soc. (3) 67 (1993), no. 1, 159–182. MR 1218124, DOI 10.1112/plms/s3-67.1.159
A. A. Il′in, Best constants for a class of polymultiplicative inequalities for derivatives, Mat. Sb. 189 (1998), no. 9, 61–84 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 9-10, 1295–1333. MR 1680856, DOI 10.1070/SM1998v189n09ABEH000349
A. A. Il′in, Lieb-Thirring integral inequalities and their applications to attractors of Navier-Stokes equations, Mat. Sb. 196 (2005), no. 1, 33–66 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 1-2, 29–61. MR 2141323, DOI 10.1070/SM2005v196n01ABEH000871
Ekkehard Krätzel, Lattice points, Mathematics and its Applications (East European Series), vol. 33, Kluwer Academic Publishers Group, Dordrecht, 1988. MR 998378
Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR 1232660, DOI 10.1007/978-3-0348-8573-7
A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), no. 2, 531–545. MR 1491551, DOI 10.1006/jfan.1997.3155
Ari Laptev and Timo Weidl, Sharp Lieb-Thirring inequalities in high dimensions, Acta Math. 184 (2000), no. 1, 87–111. MR 1756570, DOI 10.1007/BF02392782
Michael Levitin and Leonid Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002), no. 2, 425–445. MR 1923409, DOI 10.1006/jfan.2001.3913
Farouk Odeh and Joseph B. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Mathematical Phys. 5 (1964), 1499–1504. MR 168924, DOI 10.1063/1.1931182
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Robert C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977), no. 4, 525–533. MR 482597, DOI 10.1007/BF02567385
G. Rozenblum and M. Solomyak, Counting Schrödinger bound states: semiclassics and beyond, arXiv:0803.3138v2 [math.SP] 22 Nov 2008.
Yu. Safarov, Lower bounds for the generalized counting function, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 275–293. MR 1747899
M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 (Russian). MR 798454
Joachim Stubbe, Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 6, 1347–1353. MR 2734344, DOI 10.4171/JEMS/233
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
S. Wang, Generalization of the Thomas-Reiche-Kuhn and the Bethe sum rules. Phys. Rev. A 60 (1999) 262–266.
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).