Eigenvalue estimates for magnetic Schrödinger operators in domains

Frank, Rupert; Laptev, Ari; Molchanov, Stanislav

doi:10.1090/S0002-9939-08-09523-3

Eigenvalue estimates for magnetic Schrödinger operators in domains
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by Rupert L. Frank, Ari Laptev and Stanislav Molchanov PDF

Proc. Amer. Math. Soc. 136 (2008), 4245-4255

Abstract:

Inequalities are derived for sums and quotients of eigenvalues of magnetic Schrödinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.

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
Mark S. Ashbaugh and Rafael Benguria, Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials, Proc. Amer. Math. Soc. 99 (1987), no. 3, 598–599. MR 875408, DOI 10.1090/S0002-9939-1987-0875408-4
Mark S. Ashbaugh and Rafael D. Benguria, Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials, Comm. Math. Phys. 124 (1989), no. 3, 403–415. MR 1012632
Mark S. Ashbaugh and Rafael D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math. (2) 135 (1992), no. 3, 601–628. MR 1166646, DOI 10.2307/2946578
F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504
Qing-Ming Cheng and Hongcang Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann. 337 (2007), no. 1, 159–175. MR 2262780, DOI 10.1007/s00208-006-0030-x
Giuseppe Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital. A (6) 1 (1982), no. 1, 145–151 (English, with Italian summary). MR 652376
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
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
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
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, Recent results on Lieb-Thirring inequalities, Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000) Univ. Nantes, Nantes, 2000, pp. Exp. No. XX, 14. MR 1775696
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
Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
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
Barry Simon, Kato’s inequality and the comparison of semigroups, J. Functional Analysis 32 (1979), no. 1, 97–101. MR 533221, DOI 10.1016/0022-1236(79)90079-X
Barry Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1979), no. 1, 37–47. MR 526289
H. 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).