no. 2
Heat traces and existence of scattering resonances for bounded potentials
[La trace du noyau de la chaleur et l’existence de résonances de diffusion pour les potentiels bornés]
Smith, Hart F.1 ; Zworski, Maciej2
1 Department of Mathematics University of Washington Seattle, WA 98195 (USA)
2 Department of Mathematics University of California Berkeley, CA 94720 (USA)
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 455-475.

Nous montrons qu’en dimensions impaires, un potentiel borné, à support compact et à valeurs réelles, présente au moins une résonance de diffusion. En dimension 3 ou plus, ce résultat était connu seulement pour des potentiels suffisamment réguliers. La démonstration est fondée sur un résultat inverse, montrant que la trace régularisée du noyau de la chaleur associé admet un développement asymptotique complet si et seulement si le potentiel est lisse.

We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. In dimensions 3 and greater this was previously known only for sufficiently smooth potentials. The proof is based on an inverse result, which shows that the regularized trace of the associated heat kernel admits a full asymptotic expansion if and only if the potential is smooth.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3016
Classification : 35P25, 35K08
Keywords: Scattering, resonances, heat trace
Mot clés : Diffusion, résonances, noyau de la chaleur

Smith, Hart F. 1 ; Zworski, Maciej 2

1 Department of Mathematics University of Washington Seattle, WA 98195 (USA)
2 Department of Mathematics University of California Berkeley, CA 94720 (USA) 
@article{AIF_2016__66_2_455_0,
     author = {Smith, Hart F. and Zworski, Maciej},
     title = {Heat traces and existence of scattering resonances for bounded potentials},
     journal = {Annales de l'Institut Fourier},
     pages = {455--475},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     doi = {10.5802/aif.3016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3016/}
}
TY  - JOUR
AU  - Smith, Hart F.
AU  - Zworski, Maciej
TI  - Heat traces and existence of scattering resonances for bounded potentials
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 455
EP  - 475
VL  - 66
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3016/
DO  - 10.5802/aif.3016
LA  - en
ID  - AIF_2016__66_2_455_0
ER  - 
%0 Journal Article
%A Smith, Hart F.
%A Zworski, Maciej
%T Heat traces and existence of scattering resonances for bounded potentials
%J Annales de l'Institut Fourier
%D 2016
%P 455-475
%V 66
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3016/
%R 10.5802/aif.3016
%G en
%F AIF_2016__66_2_455_0
Smith, Hart F.; Zworski, Maciej. Heat traces and existence of scattering resonances for bounded potentials. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 455-475. doi : 10.5802/aif.3016. https://aif.centre-mersenne.org/articles/10.5802/aif.3016/

[1] Autin, Aymeric Isoresonant complex-valued potentials and symmetries, Canad. J. Math., Volume 63 (2011) no. 4, pp. 721-754 | DOI

[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

[3] van den Berg, M. On the trace of the difference of Schrödinger heat semigroups, Proc. Roy. Soc. Edinburgh Sect. A, Volume 119 (1991) no. 1-2, pp. 169-175 | DOI

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

[5] Christiansen, T. Some lower bounds on the number of resonances in Euclidean scattering, Math. Res. Lett., Volume 6 (1999) no. 2, pp. 203-211 | DOI

[6] Christiansen, T. Several complex variables and the distribution of resonances in potential scattering, Comm. Math. Phys., Volume 259 (2005) no. 3, pp. 711-728 | DOI

[7] Christiansen, T. Schrödinger operators with complex-valued potentials and no resonances, Duke Math. J., Volume 133 (2006) no. 2, pp. 313-323 | DOI

[8] Christiansen, T. Isophasal, isopolar, and isospectral Schrödinger operators and elementary complex analysis, Amer. J. Math., Volume 130 (2008) no. 1, pp. 49-58 | DOI

[9] Christiansen, T.; Hislop, P. D. The resonance counting function for Schrödinger operators with generic potentials, Math. Res. Lett., Volume 12 (2005) no. 5-6, pp. 821-826 | DOI

[10] Christiansen, T.; Hislop, P. D. Maximal order of growth for the resonance counting functions for generic potentials in even dimensions, Indiana Univ. Math. J., Volume 59 (2010) no. 2, pp. 621-660 | DOI

[11] Datchev, Kiril; Hezari, Hamid Resonant uniqueness of radial semiclassical Schrödinger operators, Appl. Math. Res. Express. AMRX (2012) no. 1, pp. 105-113

[12] Dinh, Tien-Cuong; Vu, Duc-Viet Asymptotic number of scattering resonances for generic Schrödinger operators, Comm. Math. Phys., Volume 326 (2014) no. 1, pp. 185-208 | DOI

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

[14] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (in preparation,http://math.berkeley.edu/~zworski/res.pdf)

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

[16] Guillopé, Laurent Asymptotique de la phase de diffusion pour l’opérateur de Schrödinger avec potentiel, C. R. Acad. Sci. Paris Sér. I Math., Volume 293 (1981) no. 12, pp. 601-603

[17] 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

[18] Jensen, Arne High energy asymptotics for the total scattering phase in potential scattering theory, Functional-analytic methods for partial differential equations (Tokyo, 1989) (Lecture Notes in Math.), Volume 1450, Springer, Berlin, 1990, pp. 187-195 | DOI

[19] Jensen, Arne; Kato, Tosio Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., Volume 46 (1979) no. 3, pp. 583-611 http://projecteuclid.org/euclid.dmj/1077313577 | DOI

[20] Korotyaev, Evgeny Inverse resonance scattering on the real line, Inverse Problems, Volume 21 (2005) no. 1, pp. 325-341 | DOI

[21] McKean, H. P.; van Moerbeke, P. The spectrum of Hill’s equation, Invent. Math., Volume 30 (1975) no. 3, pp. 217-274 | DOI

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

[23] Sá Barreto, Antonio Remarks on the distribution of resonances in odd dimensional Euclidean scattering, Asymptot. Anal., Volume 27 (2001) no. 2, pp. 161-170

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

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

[26] Titchmarsh, E. C. The theory of functions, Oxford University Press, Oxford, 1958, x+454 pages Reprint of the second (1939) edition

[27] 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

[28] Colin de Verdière, Yves Semiclassical trace formulas and heat expansions, Anal. PDE, Volume 5 (2012) no. 3, pp. 693-703 | DOI

[29] Zworski, Maciej Sharp polynomial bounds on the number of scattering poles, Duke Math. J., Volume 59 (1989) no. 2, pp. 311-323 | DOI

[30] Zworski, Maciej Poisson formulae for resonances, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytech., Palaiseau, 1997, Exp. No. XIII, 14 pages

[31] Zworski, Maciej A remark on isopolar potentials, SIAM J. Math. Anal., Volume 32 (2001) no. 6, p. 1324-1326 (electronic) | DOI

Cité par Sources :