A note on an upper bound of the toll of negative eigenvalues for a fractional Schr?dinger operator

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• 作者：Mohammed El A?di
• 关键词：Negative eigenvalues ; Variational principle ; Fourier transform ; Quadratic forms ; Primary 34B09 ; 34L15 ; 34L25 ; 34L05 ; 35J40 ; 35P15 ; 35R06 ; 35R15 ; 47A75 ; Secondary 47A07 ; 47A40 ; 47A10 ; 57R40 ; 58D10
• 刊名：Journal of Pseudo-Differential Operators and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：6
• 期：4
• 页码：567-571
• 全文大小：409 KB
• 参考文献：1.Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225-36 (2004)MATH MathSciNet CrossRef
  2.Egorov, Y.V.: Sur des estimations des valeurs propres d’opérateurs elliptiques. Séminaire équations aux dérivées partielles (dit “Goulaouic-Schwartz-, pp. 1-1 (1991-992). http://?eudml.?org/?doc/-12033
  3.Egorov, Yu. V., Kondrat’ev, V.A.: On the negative spectrum of an elliptic operator (in Russian). Mat. Sb. 181(2), 147-66 (1990). (English transl. in Math. USSR-Sb. 69(1), 155-77 (1991))
  4.Egorov, YuV, Kondratiev, V.A.: On Moments of negative eigenvalues of an elliptic operator. Math. Nachr. 174, 73-9 (1995)MATH MathSciNet CrossRef
  5.Frank, R., Lieb, E.H., Seringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schr?dinger operators. J. Am. Math. Soc. 21(4), 925-50 (2008)MATH CrossRef
  6.Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators (in Russian). Gosudarstv. Izdat. Fiz. Mat. Lit. Moscow (1963). (English translation: Daniel Davey and Co., New York (1966))
  7.Herbst, I.W.: Spectral theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\) . Commun. Math. Phys. 53(3), 285-94 (1977)MATH MathSciNet CrossRef
  8.Lieb, E.: Bounds on the eigenvalues of the Laplace and Schr?dinger operator. Bull. Am. Math. Soc. 82(5), 751-53 (1976)MATH MathSciNet CrossRef
  9.Rozenblyum, G.V.: The distribution of the discrete spectrum of singular differential operators. Izv. Vyssh. Uchebn. Zved. Mat. 1(164), 75-6 (1976). (English transl. in Soviet Math. (Iz. VUZ) 20 (1976))
  10.Rozenblum, G., Sobolev, A.V.: Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential. In: American Mathematical Society Transactions—Series 2. Advances in the Mathematical Sciences, Spectral Theory of Differential Operators M.Sh. Birman 80th Anniversary Collection, vol. 225, pp. 169-90 (2008)
  11.Ruppert, F.L.: A simple proof of Hardy–Lieb–Thirring inequalities. Commun. Math. Phys. 290(2), 789-00 (2009)CrossRef
  
• 作者单位：Mohammed El A?di (1)
  
  1. Departamento de Matemáticas, Universidad Nacional de Colombia, sede Bogotá, Avenida carrera 30 número 45-03, Edificio 404, Bogotá, DC, Colombia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：None Assigned
• 出版者：Birkh盲user Basel
• ISSN：1662-999X

文摘

For a suitable potential, we provide an upper bound of the number of negative eigenvalues corresponding to a fractional Schr?dinger operator, we observe that, up to a multiplicative constant, this upper bound is the same to the non-relativistic case. Keywords Negative eigenvalues Variational principle Fourier transform Quadratic forms