Scattering resonances for highly oscillatory potentials

Drouot, Alexis

doi:10.24033/asens.2368

Geodesic

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Scattering resonances for highly oscillatory potentials
[Résonances de potentiels rapidement oscillants]

Drouot, Alexis

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 865-925.

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Abstract (VO)
Résumé

We study resonances of compactly supported potentials $V_{ε} (x) = W (x, x / ε)$ where $W : ℝ^{d} \times ℝ^{d} / {(2 π ℤ)}^{d} \to ℂ$ , $d$ odd. That means that $V_{ε}$ is a sum of a slowly varying potential, $W_{0}$ , and one oscillating at frequency $1 / ε$ . When $W_{0} \equiv 0$ we prove that there are no resonances above the line $ℑ λ = - A ln (ε^{- 1})$ , except a simple resonance near 0 when $d = 1$ . We show that this result is optimal by constructing a one-dimensional example. This settles a conjecture of Duchêne-Vukićević-Weinstein [12]. When $W_{0} \neq 0$ and $W$ smooth we prove that resonances in fixed strips admit an expansion in powers of $ε$ . The argument provides a method for computing the coefficients of the expansion. We produce an effective potential converging uniformly to $W_{0}$ as $ε \to 0$ and whose resonances approach resonances of $V_{ε}$ modulo $O (ε^{4})$ . This improves the one-dimensional result of Duchêne, Vukićević and Weinstein and extends it to all odd dimensions.

Nous étudions les résonances de potentiels à support compact $V_{ε} (x) = W (x, x / ε)$ , où $W : ℝ^{d} \times ℝ^{d} / {(2 π ℤ)}^{d} \to ℂ$ et $d$ est impair. Ainsi, $V_{ε}$ est la somme d'un potentiel qui varie lentement $W_{0}$ et d'un potentiel qui oscille à fréquence $1 / ε$ . Quand $W_{0} \equiv 0$ nous prouvons que $V_{ε}$ n'a pas de résonances dans la zone ${ℑ λ \geq - A ln (ε^{- 1})}$ mise à part une unique résonance proche de 0 si $d = 1$ . Nous montrons par un exemple explicite que ce résultat est optimal. Cela prouve une conjecture de Duchêne-Vukićević-Weinstein [12]. Quand $W_{0} \neq 0$ et $W$ est lisse nous montrons que les resonances de $V_{ε}$ qui restent bornées lorsque $ε$ tend vers 0 admettent une expansion en puissances de $ε$ . Les arguments de la preuve permettent de calculer les coefficients de cette expansion. Nous construisons un potentiel effectif qui converge uniformément vers $W_{0}$ lorsque $ε$ tend vers 0 et dont les résonances sont à distance $O (ε^{4})$ de celles de $W_{0}$ . Cela améliore et étend les résultats de Duchêne, Vukićević et Weinstein à toutes les dimensions impaires.

Publié le : 2018-05-27

MR Zbl

DOI : 10.24033/asens.2368

Classification : 35P15, 35P25, 42B20.
Keywords: Scattering resonances, highly oscillatory potentials, asymptotic expansions.
Mots-clés : Résonances, potentiels rapidement oscillants, expansions asymptotiques.

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     author = {Drouot, Alexis},
     title = {Scattering resonances for highly oscillatory potentials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {865--925},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {4},
     year = {2018},
     doi = {10.24033/asens.2368},
     mrnumber = {3861565},
     zbl = {1408.35107},
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     url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2368/}
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Drouot, Alexis. Scattering resonances for highly oscillatory potentials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 865-925. doi : 10.24033/asens.2368. http://geodesic.mathdoc.fr/articles/10.24033/asens.2368/

Bibliographie
Cité par

Blanes, S.; Casas, F. On the convergence and optimization of the Baker-Campbell-Hausdorff formula, Linear Algebra Appl., Volume 378 (2004), pp. 135-158 (ISSN: 0024-3795) | MR | Zbl | DOI

Borisov, D. I.; Gadyl'shin, R. R. On the spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential, Teoret. Mat. Fiz., Volume 147 (2006), pp. 58-63 (ISSN: 0564-6162) | MR | Zbl | DOI

Borisov, D. I. On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential, J. Math. Sci. (N.Y.), Volume 139 (2006), pp. 6243-6322 (ISSN: 1072-3374) | MR | Zbl | DOI

Borisov, D. I. On some singular perturbations of periodic operators, Teoret. Mat. Fiz., Volume 151 (2007), pp. 207-218 (ISSN: 0564-6162) | MR | Zbl | DOI

Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. On the Lambert $W$ function, Adv. Comput. Math., Volume 5 (1996), pp. 329-359 (ISSN: 1019-7168) | MR | Zbl | DOI

Christiansen, T. Schrödinger operators with complex-valued potentials and no resonances, Duke Math. J., Volume 133 (2006), pp. 313-323 (ISSN: 0012-7094) | MR | Zbl | DOI

Dimassi, M.; Duong, A. T. Scattering and semi-classical asymptotics for periodic Schrödinger operators with oscillating decaying potential, Math. J. Okayama Univ., Volume 59 (2017), pp. 149-174 (ISSN: 0030-1566) | MR | Zbl

Dimassi, M. Semi-classical asymptotics for the Schrödinger operator with oscillating decaying potential, Canad. Math. Bull., Volume 59 (2016), pp. 734-747 (ISSN: 0008-4395) | MR | Zbl | DOI

Duchêne, V.; Raymond, N. Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials (preprint arXiv:1609.01990 ) | MR

Drouot, A. Resonances for random highly oscillatory potentials (preprint arXiv:1703.08140 ) | MR

Drouot, A. Bound states for highly oscillatory potentials in dimension 2, SIAM J. Math. Anal., Volume 50 (2018), pp. 1471-1484 | MR | Zbl

Duchêne, V.; Vukićević, I.; Weinstein, M. I. Scattering and localization properties of highly oscillatory potentials, Comm. Pure Appl. Math., Volume 67 (2014), pp. 83-128 (ISSN: 0010-3640) | MR | Zbl | DOI

Duchêne, V.; Vukićević, I.; Weinstein, M. I. Homogenized description of defect modes in periodic structures with localized defects, Commun. Math. Sci., Volume 13 (2015), pp. 777-823 (ISSN: 1539-6746) | MR | Zbl | DOI

Duchêne, V.; Weinstein, M. I. Scattering, homogenization, and interface effects for oscillatory potentials with strong singularities, Multiscale Model. Simul., Volume 9 (2011), pp. 1017-1063 (ISSN: 1540-3459) | MR | Zbl | DOI

Dyatlov, S.; Zworski, M. Mathematical theory of scattering resonances (2018) (lecture notes http://math.mit.edu/~dyatlov/res/res_20180406.pdf ) | MR

Gesztesy, F.; Latushkin, Y.; Mitrea, M.; Zinchenko, M. Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., Volume 12 (2005), pp. 443-471 (ISSN: 1061-9208) | MR | Zbl

Golowich, S. E.; Weinstein, M. I. Scattering resonances of microstructures and homogenization theory, Multiscale Model. Simul., Volume 3 (2005), pp. 477-521 (ISSN: 1540-3459) | MR | Zbl | DOI

Klopp, F., Spectral analysis of quantum Hamiltonians (Oper. Theory Adv. Appl.), Volume 224, Birkhäuser, 2012, pp. 171-182 | MR | Zbl | DOI

Klopp, F. Resonances for large one-dimensional “ergodic” systems, Anal. PDE, Volume 9 (2016), pp. 259-352 (ISSN: 2157-5045) | MR | Zbl | DOI

Phong, T. T. Resonances for 1D half-line periodic operators: I. Generic case (preprint arXiv:1509.03788 )

Phong, T. T. Resonances for 1D half-line periodic operators: II. Special case (preprint arXiv:1509.06133 )

Sá Barreto, A.; Zworski, M. Existence of resonances in potential scattering, Comm. Pure Appl. Math., Volume 49 (1996), pp. 1271-1280 (ISSN: 0010-3640) | MR | Zbl | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

Simon, B. Notes on infinite determinants of Hilbert space operators, Advances in Math., Volume 24 (1977), pp. 244-273 (ISSN: 0001-8708) | MR | Zbl | DOI

Smith, H. F.; Zworski, M. Heat traces and existence of scattering resonances for bounded potentials, Ann. Inst. Fourier (Grenoble), Volume 66 (2016), pp. 455-475 http://aif.cedram.org/item?id=AIF_2016__66_2_455_0 (ISSN: 0373-0956) | MR | Zbl | mathdoc-id | DOI

Titchmarsh, E. C., Oxford Univ. Press, Oxford, 1939, 454 pages | MR | JFM

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