Dispersive estimates and absence of embedded eigenvalues

Koch, Herbert; Tataru, Daniel

doi:10.5802/jedp.19

Koch, Herbert ¹ ; Tataru, Daniel ²

¹ Fachbereich Mathematik, Universität Dortmund
² Department of Mathematics, University of California, Berkeley

Journées équations aux dérivées partielles (2005), article no. 6, 10 p.

Résumé

In [2] Kenig, Ruiz and Sogge proved

{∥ u ∥}_{L^{\frac{2 n}{n - 2}} (ℝ^{n})} ≲ {∥ L u ∥}_{L^{\frac{2 n}{n + 2}} (ℝ^{n})}

provided $n \geq 3$ , $u \in C_{0}^{\infty} (ℝ^{n})$ and $L$ is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with $C^{2}$ coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues for potentials in $L^{\frac{n + 1}{2}}$ and variants thereof.

DOI : 10.5802/jedp.19

Affiliations des auteurs :

Koch, Herbert ¹ ; Tataru, Daniel ²

¹ Fachbereich Mathematik, Universität Dortmund
² Department of Mathematics, University of California, Berkeley

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Koch, Herbert; Tataru, Daniel. Dispersive estimates and absence of embedded eigenvalues. Journées équations aux dérivées partielles (2005), article  no. 6, 10 p. doi : 10.5802/jedp.19. http://www.numdam.org/articles/10.5802/jedp.19/

Bibliographie
Cité par

[1] A. D. Ionescu and D. Jerison. On the absence of positive eigenvalues of Schrödinger operators with rough potentials. Geom. Funct. Anal., 13(5):1029–1081, 2003. | MR | Zbl

[2] C. E. Kenig, A. Ruiz, and C. D. Sogge. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J., 55(2):329–347, 1987. | MR | Zbl

[3] Herbert Koch and Daniel Tataru. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math., 58(2):217–284, 2005. | MR | Zbl

[4] Michael Reed and Barry Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. | MR

[5] Christopher D. Sogge. Fourier integrals in classical analysis, volume 105 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993. | MR | Zbl

[6] J. von Neumann and E.P. Wigner. Über merkwürdige diskrete Eigenwerte. Z. Phys., 30:465–467, 1929.

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