Dispersive estimates and absence of embedded eigenvalues

Herbert Koch[1]; Daniel Tataru[2]

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

Journées Équations aux dérivées partielles (2005)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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In [2] Kenig, Ruiz and Sogge proved ∥ u ∥ L 2 n n - 2 ( ℝ n ) ≲ ∥ L u ∥ L 2 n n + 2 ( ℝ n ) provided n ≥ 3 , u ∈ C 0 ∞ ( 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 n + 1 2 and variants thereof.

How to cite

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Koch, Herbert, and Tataru, Daniel. "Dispersive estimates and absence of embedded eigenvalues." Journées Équations aux dérivées partielles (2005): 1-10. <http://eudml.org/doc/10613>.

@article{Koch2005,
abstract = {In [2] Kenig, Ruiz and Sogge proved\[ \Vert u \Vert \_\{L^\{\frac\{2n\}\{n-2\}\}(\mathbb\{R\}^n)\} \lesssim \Vert L u \Vert \_\{L^\{\frac\{2n\}\{n+2\}\}(\mathbb\{R\}^n)\} \]provided $n \ge 3$, $u \in C^\{\infty \}_0(\mathbb\{R\}^n)$ 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.},
affiliation = {Fachbereich Mathematik, Universität Dortmund; Department of Mathematics, University of California, Berkeley},
author = {Koch, Herbert, Tataru, Daniel},
journal = {Journées Équations aux dérivées partielles},
keywords = {Schrödinger operator; embedded eigenvalues; Carleman estimates},
language = {eng},
month = {6},
pages = {1-10},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Dispersive estimates and absence of embedded eigenvalues},
url = {http://eudml.org/doc/10613},
year = {2005},
}

TY - JOUR
AU - Koch, Herbert
AU - Tataru, Daniel
TI - Dispersive estimates and absence of embedded eigenvalues
JO - Journées Équations aux dérivées partielles
DA - 2005/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 10
AB - In [2] Kenig, Ruiz and Sogge proved\[ \Vert u \Vert _{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \lesssim \Vert L u \Vert _{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \]provided $n \ge 3$, $u \in C^{\infty }_0(\mathbb{R}^n)$ 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.
LA - eng
KW - Schrödinger operator; embedded eigenvalues; Carleman estimates
UR - http://eudml.org/doc/10613
ER -

References

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  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. Zbl1055.35098MR2024415
  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. Zbl0644.35012MR894584
  3. Herbert Koch and Daniel Tataru. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math., 58(2):217–284, 2005. Zbl1078.35143MR2094851
  4. Michael Reed and Barry Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. Zbl0401.47001MR493421
  5. Christopher D. Sogge. Fourier integrals in classical analysis, volume 105 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993. Zbl0783.35001MR1205579
  6. J. von Neumann and E.P. Wigner. Über merkwürdige diskrete Eigenwerte. Z. Phys., 30:465–467, 1929. Zbl55.0520.04

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