Explicit formulas for the Schrödinger wave operators in $ {\mathbb{R}}^{2}$

Serge Richard; Rafael Tiedra de Aldecoa

doi:10.1016/j.crma.2013.03.006

Functional Analysis/Mathematical Physics

Explicit formulas for the Schrödinger wave operators in

R^{2}

Presented by: Jean-Michel Bony

Serge Richard ¹; Rafael Tiedra de Aldecoa ²

¹ Université de Lyon, université Lyon-1, CNRS, UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France
² Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 209-214.

Abstract
Résumé

In this note, we derive explicit formulas for the Schrödinger wave operators in $R^{2}$ under the assumption that the 0-energy is neither an eigenvalue nor a resonance. These formulas justify the use of a recently introduced topological approach of scattering theory to obtain index theorems.

Dans cette note, nous dérivons des formules explicites pour les opérateurs dʼonde de Schrödinger dans $R^{2}$ , sous lʼhypothèse que lʼénergie 0 nʼest, ni une valeur propre, ni une résonance. Ces formules légitiment lʼemploi dʼune approche topologique de la théorie de la diffusion récemment introduite pour obtenir des théorèmes dʼindice.

Received: 2012-12-10
Accepted: 2013-03-15
Published online: 2013-03-01

DOI: 10.1016/j.crma.2013.03.006

Author's affiliations:

Serge Richard ¹; Rafael Tiedra de Aldecoa ²

¹ Université de Lyon, université Lyon-1, CNRS, UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France
² Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

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     author = {Serge Richard and Rafael Tiedra de Aldecoa},
     title = {Explicit formulas for the {Schr\"odinger} wave operators in $ {\mathbb{R}}^{2}$},
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Serge Richard; Rafael Tiedra de Aldecoa. Explicit formulas for the Schrödinger wave operators in $ {\mathbb{R}}^{2}$. Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 209-214. doi : 10.1016/j.crma.2013.03.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.03.006/

References
Cited by

[1] S. Agmon Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 2 (1975) no. 2, pp. 151-218

[2] W.O. Amrein; A. Boutet de Monvel; V. Georgescu $C_{0}$ -groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Prog. Math., vol. 135, Birkhäuser, Basel, 1996

[3] J. Bellissard; H. Schulz-Baldes Scattering theory for lattice operators in dimension $d ⩾ 3$ , Rev. Math. Phys., Volume 24 (2012) no. 08, p. 1250020

[4] D. Bollé; F. Gesztesy; C. Danneels Threshold scattering in two dimensions, Ann. Inst. Henri Poincaré, a Phys. Théor., Volume 48 (1988) no. 2, pp. 175-204

[5] M.B. Erdoğan, W.R. Green, A weighted dispersive estimate for Schrödinger operators in dimension two, Commun. Math. Phys., , in press, preprint on . | arXiv | DOI

[6] M.B. Erdoğan, W.R. Green, Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy, Trans. Amer. Math. Soc., in press, preprint on . | arXiv

[7] H. Isozaki; S. Richard On the wave operators for the Friedrichs–Faddeev model, Ann. Inst. Henri Poincaré, Volume 13 (2012), pp. 1469-1482

[8] A. Jeffrey Handbook of Mathematical Formulas and Integrals, Academic Press, Inc., San Diego, CA, 1995

[9] A. Jensen; G. Nenciu A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., Volume 13 (2001) no. 6, pp. 717-754

[10] A. Jensen; K. Yajima A remark on $L^{p}$ -boundedness of wave operators for two-dimensional Schrödinger operators, Commun. Math. Phys., Volume 225 (2002) no. 3, pp. 633-637

[11] T. Kato Growth properties of solutions of the reduced wave equation with a variable coefficient, Commun. Pure Appl. Math., Volume 12 (1959), pp. 403-425

[12] J. Kellendonk; K. Pankrashkin; S. Richard Levinsonʼs theorem and higher degree traces for the Aharonov–Bohm operators, J. Math. Phys., Volume 52 (2011), p. 052102

[13] J. Kellendonk; S. Richard Levinsonʼs theorem for Schrödinger operators with point interaction: a topological approach, J. Phys. A, Volume 39 (2006) no. 46, pp. 14397-14403

[14] J. Kellendonk; S. Richard On the structure of the wave operators in one dimensional potential scattering, Math. Phys. Electron. J., Volume 14 (2008), pp. 1-21

[15] J. Kellendonk; S. Richard On the wave operators and Levinsonʼs theorem for potential scattering in $R^{3}$ , Asian-Eur. J. Math., Volume 5 (2012), p. 1250004-1-1250004-22

[16] N.N. Khuri; A. Martin; J.-M. Richard; T.T. Wu Universality of low-energy scattering in $2 + 1$ dimensions: the nonsymmetric case, J. Math. Phys., Volume 46 (2005) no. 3, p. 032103

[17] N.N. Khuri; A. Martin; J.-M. Richard; T.T. Wu Low-energy potential scattering in two and three dimensions, J. Math. Phys., Volume 50 (2009) no. 7, p. 072105

[18] S.T. Kuroda Scattering theory for differential operators. I. Operator theory, J. Math. Soc. Jpn., Volume 25 (1973), pp. 75-104

[19] K. Pankrashkin; S. Richard Spectral and scattering theory for the Aharonov–Bohm operators, Rev. Math. Phys., Volume 23 (2011), pp. 53-81

[20] S. Richard; R. Tiedra de Aldecoa New formulae for the wave operators for a rank one interaction, Integral Equations Operator Theory, Volume 66 (2010), pp. 283-292

[21] S. Richard; R. Tiedra de Aldecoa New expressions for the wave operators of Schrödinger operators in $R^{3}$ (preprint on) | arXiv

[22] W. Schlag Dispersive estimates for Schrödinger operators in dimension two, Commun. Math. Phys., Volume 257 (2005) no. 1, pp. 87-117

[23] R. Weder Universality of entanglement creation in low-energy two-dimensional scattering (preprint on) | arXiv

[24] D.R. Yafaev Mathematical Scattering Theory, Transl. Math. Monogr., vol. 105, American Mathematical Society, Providence, RI, 1992

[25] D.R. Yafaev Mathematical Scattering Theory. Analytic Theory, Math. Surveys Monogr., vol. 158, American Mathematical Society, Providence, RI, 2010

[26] K. Yajima $L^{p}$ -boundedness of wave operators for two-dimensional Schrödinger operators, Commun. Math. Phys., Volume 208 (1999) no. 1, pp. 125-152

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