Abstract
The classical Hardy inequality for the Laplacian Δ on \({\mathbb{R}^n}\) shows the borderline-behavior of a potential V for the following question: whether the Schrödinger operator −Δ + V has a finite or infinite number of the discrete spectrum. In this paper, we will give a sharp generalization of this inequality on \({\mathbb{R}^n}\) to a relative version of that on large classes of complete noncompact manifolds. Replacing \({\mathbb{R}^n}\) by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some sharp criteria for the above-type question.
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References
Andersson L.: Elliptic systems on manifolds with asymptotically negative curvature. Indiana Univ. Math. J. 42, 1359–1388 (1993)
Bando S., Kasue A., Nakajima H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97, 313–349 (1989)
Barta J.: Sur la vibration fondamental d’une menbrane. Comptes rendus Acad. Sci. Paris 204, 472–473 (1937)
Carron G.: Inégalités de Hardy sur les variétés riemanniennes non-compactes. J. Math. Pures Appl. 76(9), 883–891 (1997)
Chavel, I.: Eigenvalues in Riemannian Geometry, Pure and Applied Mathamatics. Academic Press, New York (1984)
Davies E.B.: Some norm bounds and quadratic form inequalities for Schrödinger operators II. J. Oper. Theory 12, 177–196 (1984)
Davies E.B.: A review of Hardy inequalities. Oper. Theory Adv. Appl. 110, 55–67 (1998)
Fabes E., Kenig C., Serapioni R.: The local regularity of solutions of degenerate elliptic equation. Commun. Partial Differ. Equ. 7, 77–116 (1982)
Graham C.R., Lee J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)
Greene, R.E., Wu, H.: Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Mathamatics, vol. 699. Springer, New York (1979)
Gustafson S.J., Sigal I.M.: Mathematical Concepts of Quantum Mechanics. Springer, New York (2003)
Hardy G.H., Littlewood J.E., Olya G.P.: Inequalities. Cambridge University Press, Cambridge (1952)
Hassell A., Marshall S.: Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree −2. Trans. Am. Math. Soc. 360, 4145–4167 (2008)
Heintze E., Karcher H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. Ecole Norm. Sup. Paris 11, 451–470 (1978)
Kasue A.: On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Ann. Sci. Ecole Norm. Sup. 17, 31–44 (1984)
Kirsch W., Simon B.: Corrections to the classical behavior of the number of bound states of Schrödinger operators. Ann. Phys. 183, 122–130 (1988)
Kombe I., Özaydin M.: Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 361(12), 6191–6203 (2009)
Kombe, I., Özaydin, M.: Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds. arXiv:1103.2747 (preprint)
Kumura H.: On the essential spectrum of the Laplacian on complete manifolds. J. Math. Soc. Jpn. 49, 1–14 (1997)
Lee, J.M.: Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds. Memoirs of the American Mathematical Society, vol. 864. American Mathematical Society (2006)
Levin D., Solomyak M.: The Rozenblum–Lieb–Cwikel inequality for Markov generators. J. Anal. Math. 71, 173–193 (1997)
Mazzeo R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28, 309–339 (1988)
Murata M.: Finiteness of negative spectra of elliptic operators. Can. Math. Bull. 28, 487–491 (1985)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. II. Academic Press, New York (1972)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV, Academic Press, New York (1978)
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Communicated by Y. Giga.
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Akutagawa, K., Kumura, H. Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds. Calc. Var. 48, 67–88 (2013). https://doi.org/10.1007/s00526-012-0542-z
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DOI: https://doi.org/10.1007/s00526-012-0542-z