We prove the analog of the Cwikel-Lieb-Rozenblum estimate for a wide class of second-order elliptic operators by two different tools: Lieb-Thirring inequalities for Schrödinger operators with matrix-valued potentials and Sobolev inequalities for warped product spaces.
@article{bwmeta1.element.doi-10_1007_s11533-005-0008-z,
author = {Daniel Levin},
title = {On some new spectral estimates for Schr\"odinger-like operators},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {123-137},
zbl = {1128.35076},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0008-z}
}
Daniel Levin. On some new spectral estimates for Schrödinger-like operators. Open Mathematics, Tome 4 (2006) pp. 123-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0008-z/
[1] W. Beckner: “On the Grushin operator and hyperbolic symmetry”, Proc. Amer. Math. Soc., Vol. 129, (2001), pp. 1233–1246. http://dx.doi.org/10.1090/S0002-9939-00-05630-6 | Zbl 0987.47037
[2] A. Benedek and R. Panzone: “The spaces L p with mixed norms”, Duke Math. J., Vol. 28, (1961), pp. 301–324. http://dx.doi.org/10.1215/S0012-7094-61-02828-9 | Zbl 0107.08902
[3] M. Birman and M. Solomyak: “Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory”, Amer. Math. Soc. Transl. 2 Vol. 114, (1980).
[4] T. Coulhon, A. Grigor'yan and D. Levin: “On isoperimetric dimensions of product spaces”, Commun. Anal. Geom., Vol. 11, (2003), pp. 85–120. | Zbl 1085.53027
[5] D. Hundertmark: “On the number of bound states for Schrödinger operators with operator-valued potentials”, Arkiv för matematik, Vol. 40, (2002), pp. 73–87. http://dx.doi.org/10.1007/BF02384503
[6] D. Levin and M. Solomyak: “The Rozenblum-Lieb-Cwikel inequality for Markov generators”, J. d'Analyse Math., Vol. 71, (1997), pp. 173–193. http://dx.doi.org/10.1007/BF02788029 | Zbl 0910.47017
[7] P. Maheux and L. Saloff-Coste: “Analyse sur les boules d'un opérateur sous-elliptique” (French) [Analysis on the balls of a subelliptic operator], Math. Ann., Vol. 303, (1995), pp. 713–740. http://dx.doi.org/10.1007/BF01461013 | Zbl 0836.35106
[8] M. Melgaard and G. Rozenblum: “Spectral estimates for magnetic operators”, Math. Scand., Vol. 79, (1996), pp. 237–254. | Zbl 0888.35074
[9] M. Melgaard, E.-M. Ouhabaz and G. Rozenblum: “Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians”, Ann. Henri Poincaré, Vol. 5, (2004), pp. 979–1012. http://dx.doi.org/10.1007/s00023-004-0187-3 | Zbl 1059.81049
[10] G.V. Rozenbljum: “Distribution of the discrete spectrum of singular differential operators”, Dokl. Akad. Nauk SSSR, Vol. 202, (1972), pp. 1012–1015.
[11] G. Rozenblum and M. Solomyak: “CLR-estimate revisited: Lieb's approach with no path integrals”, In: Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997), Vol. XVI, École Polytech., Palaiseau, 1997, pp. 10. | Zbl 1027.47020
[12] M. Solomyak: “Piecewise-polynomial approximation of functions from H l((0, 1)d), 2l = d, and applications to the spectral theory of the Schrödinger operator”, Israel J. Math., Vol. 86, (1994), pp. 253–275.
[13] K. Tachizawa: “On the moments of the negative eigenvalues of elliptic operators”, J. Fourier Anal. Appl., Vol. 8, (2002), pp. 233–244. http://dx.doi.org/10.1007/s00041-002-0010-9 | Zbl 1038.35045
[14] K. Tachizawa: “A generalization of the Lieb-Thirring inequalities in low dimensions”, Hokkaido Math. J., Vol. 32, (2003), pp. 383–399. | Zbl 1054.35043
[15] G.M. Tashchiyan: “The classical formula of the asymptotic behavior of the spectrum of elliptic equations that are degenerate on the boundary of the domain”, Mat. Zametki, Vol. 30, (1981), pp. 871–880, 959; English translation: Math. Notes, Vol. 30, (1981), pp. 937-942. | Zbl 0482.35067
[16] G.M. Tashchiyan: “On the distribution of eigenvalues of the elliptic Dirichlet problem”, Vestnik Leningrad. Univ. Math., Vol. 7; Mat. Meh. Astronom., Vol. 2, (1975), pp. 58–62, 171; English translation: Vestnik Leningrad. Univ. Math., Vol. 8, (1980), pp. 233–238.