Semiclassical resolvent estimates at trapped sets

Kiril Datchev[1]; András Vasy[2]

  • [1] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4397, U.S.A.
  • [2] Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2379-2384
  • ISSN: 0373-0956

Abstract

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We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs χ microlocally supported away from the trapping: ∥ χ R h ( E + i 0 ) χ ∥ = 𝒪 ( h - 1 ) , a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜ , to be supported at the trapped set, giving ∥ χ R h ( E + i 0 ) χ ˜ ∥ = 𝒪 ( a ( h ) h - 1 ) when the a priori bound is ∥ χ ˜ R h ( E + i 0 ) χ ˜ ∥ = 𝒪 ( a ( h ) h - 1 ) .

How to cite

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Datchev, Kiril, and Vasy, András. "Semiclassical resolvent estimates at trapped sets." Annales de l’institut Fourier 62.6 (2012): 2379-2384. <http://eudml.org/doc/251071>.

@article{Datchev2012,
abstract = {We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $\chi $ microlocally supported away from the trapping: $\Vert \chi R_h(E+i0)\chi \Vert = \mathcal\{O\}(h^\{-1\})$, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde\{\chi \}$, to be supported at the trapped set, giving $\Vert \chi R_h(E+i0)\tilde\{\chi \}\Vert = \mathcal\{O\}(\sqrt\{a(h)\}h^\{-1\})$ when the a priori bound is $\Vert \tilde\{\chi \}R_h(E+i0)\tilde\{\chi \}\Vert = \mathcal\{O\}(a(h)h^\{-1\})$.},
affiliation = {Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4397, U.S.A.; Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.},
author = {Datchev, Kiril, Vasy, András},
journal = {Annales de l’institut Fourier},
keywords = {Resolvent estimates; trapping; propagation of singularities; resolvent estimates},
language = {eng},
number = {6},
pages = {2379-2384},
publisher = {Association des Annales de l’institut Fourier},
title = {Semiclassical resolvent estimates at trapped sets},
url = {http://eudml.org/doc/251071},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Datchev, Kiril
AU - Vasy, András
TI - Semiclassical resolvent estimates at trapped sets
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2379
EP - 2384
AB - We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $\chi $ microlocally supported away from the trapping: $\Vert \chi R_h(E+i0)\chi \Vert = \mathcal{O}(h^{-1})$, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{\chi }$, to be supported at the trapped set, giving $\Vert \chi R_h(E+i0)\tilde{\chi }\Vert = \mathcal{O}(\sqrt{a(h)}h^{-1})$ when the a priori bound is $\Vert \tilde{\chi }R_h(E+i0)\tilde{\chi }\Vert = \mathcal{O}(a(h)h^{-1})$.
LA - eng
KW - Resolvent estimates; trapping; propagation of singularities; resolvent estimates
UR - http://eudml.org/doc/251071
ER -

References

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  1. Nicolas Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math. 124 (2002), 677-755 Zbl1013.35019MR1914456
  2. Nicolas Burq, Maciej Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17:2 (2004), 443-471 Zbl1050.35058MR2051618
  3. Fernando Cardoso, Georgi Vodev, Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II, Ann. Henri Poincaré 3 (2002), 673-691 Zbl1021.58016MR1933365
  4. Hans Christianson, Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal. 246 (2007), 145-195 Zbl1119.58018MR2321040
  5. Hans Christianson, Emmanuel Schenck, András Vasy, Jared Wunsch, From resolvent estimates to damped waves 
  6. Kiril Datchev, András Vasy, Propagation through trapped sets and semiclassical resolvent estimates, Annales de l’Institut Fourier 62.6 (2012), 2345-2375 Zbl1271.58014
  7. Lars Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, (1994), Springer Verlag Zbl0601.35001MR1313500

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