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2006 | 4 | 1 | 123-137
Tytuł artykułu

On some new spectral estimates for Schrödinger-like operators

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
123-137
Opis fizyczny
Daty
wydano
2006-03-01
online
2006-03-01
Twórcy
autor
Bibliografia
  • [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
  • [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
  • [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.
  • [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
  • [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
  • [8] M. Melgaard and G. Rozenblum: “Spectral estimates for magnetic operators”, Math. Scand., Vol. 79, (1996), pp. 237–254.
  • [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
  • [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.
  • [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
  • [14] K. Tachizawa: “A generalization of the Lieb-Thirring inequalities in low dimensions”, Hokkaido Math. J., Vol. 32, (2003), pp. 383–399.
  • [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.
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_1007_s11533-005-0008-z
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