no. 4
On the trace of the wave group and regularity of potentials
[Sur la trace du propagateur des ondes et régularité du potentiel]
Smith, Hart F.1
1 Department of Mathematics University of Washington Seattle, WA 98195-4350 (USA)
Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1453-1488.

Nous considérons l’équation des ondes avec un potentiel borné, à support compact et à valeurs réelles, et montrons que la trace régularisée de l’opérateur d’évolution associé admet un développement asymptotique à l’ordre m+2 si et seulement si le potentiel appartient à l’espace de Sobolev d’ordre m.

We consider the wave equation with a compactly supported, real-valued bounded potential, and show that the relative trace of the associated evolution group admits an asymptotic expansion to order m+2 if and only if the potential belongs to the Sobolev space of order m.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3538
Classification : 58J45, 58J50, 46E35
Keywords: Trace, wave equation with potential.
Mot clés : Trace, équation des ondes avec potentiel.

Smith, Hart F. 1

1 Department of Mathematics University of Washington Seattle, WA 98195-4350 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2023__73_4_1453_0,
     author = {Smith, Hart F.},
     title = {On the trace of the wave group and regularity of potentials},
     journal = {Annales de l'Institut Fourier},
     pages = {1453--1488},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {4},
     year = {2023},
     doi = {10.5802/aif.3538},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3538/}
}
TY  - JOUR
AU  - Smith, Hart F.
TI  - On the trace of the wave group and regularity of potentials
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 1453
EP  - 1488
VL  - 73
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3538/
DO  - 10.5802/aif.3538
LA  - en
ID  - AIF_2023__73_4_1453_0
ER  - 
%0 Journal Article
%A Smith, Hart F.
%T On the trace of the wave group and regularity of potentials
%J Annales de l'Institut Fourier
%D 2023
%P 1453-1488
%V 73
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3538/
%R 10.5802/aif.3538
%G en
%F AIF_2023__73_4_1453_0
Smith, Hart F. On the trace of the wave group and regularity of potentials. Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1453-1488. doi : 10.5802/aif.3538. https://aif.centre-mersenne.org/articles/10.5802/aif.3538/

[1] Abramowitz, Milton; Stegun, Irene A. Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964, xiv+1046 pages | MR | Zbl

[2] Bañuelos, Rodrigo; Sá Barreto, Antônio On the heat trace of Schrödinger operators, Comm. Partial Differential Equations, Volume 20 (1995) no. 11-12, pp. 2153-2164 | DOI | MR | Zbl

[3] Brüning, Jochen On the compactness of isospectral potentials, Comm. Partial Differential Equations, Volume 9 (1984) no. 7, pp. 687-698 | DOI | MR | Zbl

[4] Christiansen, T. J. Resonant rigidity for Schrödinger operators in even dimensions, Ann. Henri Poincaré, Volume 20 (2019) no. 5, pp. 1543-1582 | DOI | MR | Zbl

[5] Donnelly, Harold Compactness of isospectral potentials, Trans. Amer. Math. Soc., Volume 357 (2005) no. 5, p. 1717-1730 (electronic) | DOI | MR | Zbl

[6] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances, Graduate Studies in Mathematics, 200, American Mathematical Society, Providence, RI, 2019, xi+634 pages | DOI | MR | Zbl

[7] Fan, Ky Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A., Volume 37 (1951), pp. 760-766 | DOI | MR | Zbl

[8] Gilkey, Peter B. Asymptotic formulae in spectral geometry, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004, viii+304 pages | MR | Zbl

[9] Hislop, Peter D.; Wolf, Robert Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three, Trans. Amer. Math. Soc., Volume 374 (2021) no. 6, pp. 4481-4499 | DOI | MR | Zbl

[10] Hitrik, Michael; Polterovich, Iosif Regularized traces and Taylor expansions for the heat semigroup, J. London Math. Soc. (2), Volume 68 (2003) no. 2, pp. 402-418 | DOI | MR | Zbl

[11] Melrose, Richard Scattering theory and the trace of the wave group, J. Functional Analysis, Volume 45 (1982) no. 1, pp. 29-40 | DOI | MR | Zbl

[12] Melrose, Richard B. Geometric scattering theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995, xii+116 pages | MR | Zbl

[13] Sá Barreto, Antônio; Zworski, Maciej Existence of resonances in potential scattering, Comm. Pure Appl. Math., Volume 49 (1996) no. 12, pp. 1271-1280 | MR | Zbl

[14] Smith, Hart On the trace of Schrödinger heat kernels and regularity of potentials, Trans. Amer. Math. Soc., Volume 371 (2019) no. 6, pp. 3857-3875 | DOI | MR | Zbl

[15] Smith, Hart F.; Zworski, Maciej Heat traces and existence of scattering resonances for bounded potentials, Ann. Inst. Fourier (Grenoble), Volume 66 (2016) no. 2, pp. 455-475 | DOI | MR | Zbl

[16] Taylor, Michael E. Partial differential equations III. Nonlinear equations, Applied Mathematical Sciences, 117, Springer, New York, 2011, xxii+715 pages | DOI | MR | Zbl

[17] Colin de Verdière, Yves Une formule de traces pour l’opérateur de Schrödinger dans R 3 , Ann. Sci. École Norm. Sup. (4), Volume 14 (1981) no. 1, pp. 27-39 | DOI | MR | Zbl

[18] Watson, G. N. A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995, viii+804 pages Reprint of the second (1944) edition | MR | Zbl

Cité par Sources :