Abstract
The purpose of the present manuscript is to provide a lower bound for the number of negative eigenvalues associated to a generalized Schrödinger operator, this lower bound is given in terms of a finite number of cubes.
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El Aïdi, M.: Sur le nombre des valeurs propres négatives d’un opérateur elliptique. Bull. Sci. math. 137, 434–456 (2013)
El Aïdi, M.: Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé. Annales mathématiques Blaise Pascal 19, 197–211 (2012)
Maz’ya, V.G.: Sobolev space with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, Vol. 342, 2nd augmented Edition (2011)
Verbitsky, I.: Nonlinear potentials and trace inequalities, The Maz’ya anniversary collection, Vol. 2 (Rostock, 1998). Oper. Theory Adv. Appl. 110: 323–343 (1999)
Egorov, Y., El Aïdi, M.: Spectre négatif d’un opérateur elliptique avec des conditions au bords de Robin. Publ. Mat. 45, 125–148 (2001)
Egorov, Y.V., Kondratiev, V.A.: Estimates of the negative spectrum of an elliptic operator, in Spectral theory of operators (Novgorod, 1989). Am. Math. Soc. Transl. Ser. 2 150, 111–140 (1989)
Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106(2), 93–100 (1977)
Lieb, E.H.: Bounds on the eigenvalues of the Laplace and Schrdinger operators. Bull. Am. Math. Soc. 82(5), 751–753 (1976)
Rozenbljum, G.V.: Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk. SSSR 202, 1012–1015 (1972) (in Russian)
El Aïdi, M.: On a new embedding theorem and the CLR-type inequality for Euclidean and hyperbolic spaces. Bull. Sci. Math. 138, 335–342 (2014)
Kerman et, R., Sawyer, E.T.: The trace inequality and eigenvalue estimates for Shrödinger operators. Ann. Inst. Fourier Grenoble 4(36), 207–228 (1986)
Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)
Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics 218. Springer, Berlin (2003)
Simon, B., Reed, M.: Methods of Mathematical Physics I: Functional Analysis. Academic Press, Inc., New York (1978)
Glazman, I.M.: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Gosudarstv. Izdat. Fiz.-Mat. Lit. Moscow (1963) (English translation: Daniel Davey and Co., New York, 1966)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin (2001)
Stein, E.M.: Singular Integrals Differentiability Properties of Functions. Princeton University Press, USA (1970)
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der mathematischen Wissenschaften, Vol. 314, Springer, Berlin (1999) (corrected second printing)
Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer, Berlin (1989)
Jürgen, J.: Riemannian Geometry and Geometric Analysis, 4th edn. Springer, Berlin (2005)
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El Aïdi, M. A lower bound for the number of negative eigenvalues on a Euclidean space and on a complete Riemannian manifold. J. Pseudo-Differ. Oper. Appl. 5, 481–490 (2014). https://doi.org/10.1007/s11868-014-0098-0
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DOI: https://doi.org/10.1007/s11868-014-0098-0