Dans cette note, nous dérivons des formules explicites pour les opérateurs dʼonde de Schrödinger dans , 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.
In this note, we derive explicit formulas for the Schrödinger wave operators in 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.
Accepté le :
Publié le :
Serge Richard 1 ; Rafael Tiedra de Aldecoa 2
@article{CRMATH_2013__351_5-6_209_0, author = {Serge Richard and Rafael Tiedra de Aldecoa}, title = {Explicit formulas for the {Schr\"odinger} wave operators in $ {\mathbb{R}}^{2}$}, journal = {Comptes Rendus. Math\'ematique}, pages = {209--214}, publisher = {Elsevier}, volume = {351}, number = {5-6}, year = {2013}, doi = {10.1016/j.crma.2013.03.006}, language = {en}, }
TY - JOUR AU - Serge Richard AU - Rafael Tiedra de Aldecoa TI - Explicit formulas for the Schrödinger wave operators in $ {\mathbb{R}}^{2}$ JO - Comptes Rendus. Mathématique PY - 2013 SP - 209 EP - 214 VL - 351 IS - 5-6 PB - Elsevier DO - 10.1016/j.crma.2013.03.006 LA - en ID - CRMATH_2013__351_5-6_209_0 ER -
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/
[1] 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] -groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Prog. Math., vol. 135, Birkhäuser, Basel, 1996
[3] Scattering theory for lattice operators in dimension , Rev. Math. Phys., Volume 24 (2012) no. 08, p. 1250020
[4] 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] On the wave operators for the Friedrichs–Faddeev model, Ann. Inst. Henri Poincaré, Volume 13 (2012), pp. 1469-1482
[8] Handbook of Mathematical Formulas and Integrals, Academic Press, Inc., San Diego, CA, 1995
[9] A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., Volume 13 (2001) no. 6, pp. 717-754
[10] A remark on -boundedness of wave operators for two-dimensional Schrödinger operators, Commun. Math. Phys., Volume 225 (2002) no. 3, pp. 633-637
[11] Growth properties of solutions of the reduced wave equation with a variable coefficient, Commun. Pure Appl. Math., Volume 12 (1959), pp. 403-425
[12] Levinsonʼs theorem and higher degree traces for the Aharonov–Bohm operators, J. Math. Phys., Volume 52 (2011), p. 052102
[13] Levinsonʼs theorem for Schrödinger operators with point interaction: a topological approach, J. Phys. A, Volume 39 (2006) no. 46, pp. 14397-14403
[14] On the structure of the wave operators in one dimensional potential scattering, Math. Phys. Electron. J., Volume 14 (2008), pp. 1-21
[15] On the wave operators and Levinsonʼs theorem for potential scattering in , Asian-Eur. J. Math., Volume 5 (2012), p. 1250004-1-1250004-22
[16] Universality of low-energy scattering in dimensions: the nonsymmetric case, J. Math. Phys., Volume 46 (2005) no. 3, p. 032103
[17] Low-energy potential scattering in two and three dimensions, J. Math. Phys., Volume 50 (2009) no. 7, p. 072105
[18] Scattering theory for differential operators. I. Operator theory, J. Math. Soc. Jpn., Volume 25 (1973), pp. 75-104
[19] Spectral and scattering theory for the Aharonov–Bohm operators, Rev. Math. Phys., Volume 23 (2011), pp. 53-81
[20] New formulae for the wave operators for a rank one interaction, Integral Equations Operator Theory, Volume 66 (2010), pp. 283-292
[21] New expressions for the wave operators of Schrödinger operators in (preprint on) | arXiv
[22] Dispersive estimates for Schrödinger operators in dimension two, Commun. Math. Phys., Volume 257 (2005) no. 1, pp. 87-117
[23] Universality of entanglement creation in low-energy two-dimensional scattering (preprint on) | arXiv
[24] Mathematical Scattering Theory, Transl. Math. Monogr., vol. 105, American Mathematical Society, Providence, RI, 1992
[25] Mathematical Scattering Theory. Analytic Theory, Math. Surveys Monogr., vol. 158, American Mathematical Society, Providence, RI, 2010
[26] -boundedness of wave operators for two-dimensional Schrödinger operators, Commun. Math. Phys., Volume 208 (1999) no. 1, pp. 125-152
Cité par Sources :
Commentaires - Politique