Dispersive estimates and absence of embedded eigenvalues

Koch, Herbert; Tataru, Daniel

Koch, Herbert ; Tataru, Daniel

Journées équations aux dérivées partielles, (2005), p. 1-10 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2005-01-01

MR 2352775

DOI : https://doi.org/10.5802/jedp.19

@article{JEDP_2005____A6_0,
     author = {Koch, Herbert and Tataru, Daniel},
     title = {Dispersive estimates and absence of embedded eigenvalues},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2005},
     pages = {1-10},
     doi = {10.5802/jedp.19},
     mrnumber = {2352775},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2005____A6_0}
}

Koch, Herbert; Tataru, Daniel. Dispersive estimates and absence of embedded eigenvalues. Journées équations aux dérivées partielles,  (2005), pp. 1-10. doi : 10.5802/jedp.19. http://gdmltest.u-ga.fr/item/JEDP_2005____A6_0/

Bibliographie

[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 2024415 | Zbl 1055.35098

[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 894584 | Zbl 0644.35012

[3] Herbert Koch and Daniel Tataru. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math., 58(2):217–284, 2005. | MR 2094851 | Zbl 1078.35143

[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 493421

[5] Christopher D. Sogge. Fourier integrals in classical analysis, volume 105 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993. | MR 1205579 | Zbl 0783.35001

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