Zero energy scattering for one-dimensional Schrödinger operators and applications to dispersive estimates
HTML articles powered by AMS MathViewer
- by Iryna Egorova, Markus Holzleitner and Gerald Teschl HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 2 (2015), 51-59
Abstract:
We show that for a one-dimensional Schrödinger operator with a potential, whose $(j+1)$-th moment is integrable, the $j$-th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.References
- Tuncay Aktosun and Martin Klaus, Small-energy asymptotics for the Schrödinger equation on the line, Inverse Problems 17 (2001), no. 4, 619–632. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000). MR 1861473, DOI 10.1088/0266-5611/17/4/304
- D. Bollé, F. Gesztesy, and M. Klaus, Scattering theory for one-dimensional systems with $\int dx\,V(x)=0$, J. Math. Anal. Appl. 122 (1987), no. 2, 496–518. MR 877834, DOI 10.1016/0022-247X(87)90281-2
- D. Bollé, F. Gesztesy, and S. F. J. Wilk, A complete treatment of low-energy scattering in one dimension, J. Operator Theory 13 (1985), no. 1, 3–31. MR 768299
- Vladimir S. Buslaev and Catherine Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 3, 419–475 (English, with English and French summaries). MR 1972870, DOI 10.1016/S0294-1449(02)00018-5
- Ovidiu Costin, Wilhelm Schlag, Wolfgang Staubach, and Saleh Tanveer, Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials, J. Funct. Anal. 255 (2008), no. 9, 2321–2362. MR 2473260, DOI 10.1016/j.jfa.2008.07.015
- P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
- Iryna Egorova, Markus Holzleitner, and Gerald Teschl, Properties of the scattering matrix and dispersion estimates for Jacobi operators, J. Math. Anal. Appl. 434 (2016), no. 1, 956–966. MR 3404595, DOI 10.1016/j.jmaa.2015.09.047
- I. Egorova, E. Kopylova, and G. Teschl, Dispersion estimates for one-dimensional discrete Schrödinger and wave equations, J. Spectr. Theory 5 (2015), 663–696.
- I. Egorova, E. Kopylova, V. Marchenko, and G. Teschl, Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited, Russian Math. Surveys (to appear), arXiv:1411.0021
- Michael Goldberg, Transport in the one-dimensional Schrödinger equation, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3171–3179. MR 2322747, DOI 10.1090/S0002-9939-07-08897-1
- Katrin Grunert and Gerald Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom. 12 (2009), no. 3, 287–324. MR 2525595, DOI 10.1007/s11040-009-9062-2
- I. M. Guseĭnov, Continuity of the coefficient of reflection of a one-dimensional Schrödinger equation, Differentsial′nye Uravneniya 21 (1985), no. 11, 1993–1995, 2023 (Russian). MR 818581
- Martin Klaus, Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line, Inverse Problems 4 (1988), no. 2, 505–512. MR 954906, DOI 10.1088/0266-5611/4/2/013
- E. Kopylova and A. I. Komech, On asymptotic stability of kink for relativistic Ginzburg-Landau equations, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 213–245. MR 2835867, DOI 10.1007/s00205-011-0415-1
- Vladimir A. Marchenko, Sturm-Liouville operators and applications, Revised edition, AMS Chelsea Publishing, Providence, RI, 2011. MR 2798059, DOI 10.1090/chel/373
- Gerald Teschl, Mathematical methods in quantum mechanics, 2nd ed., Graduate Studies in Mathematics, vol. 157, American Mathematical Society, Providence, RI, 2014. With applications to Schrödinger operators. MR 3243083, DOI 10.1090/gsm/157
- Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. MR 1503035, DOI 10.2307/1968102
Additional Information
- Iryna Egorova
- Affiliation: B. Verkin Institute for Low Temperature Physics, 47, Lenin ave, 61103 Kharkiv, Ukraine
- MR Author ID: 213624
- Email: iraegorova@gmail.com
- Markus Holzleitner
- Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
- Email: amhang1@gmx.at
- Gerald Teschl
- Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria — and — International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
- MR Author ID: 360802
- Email: Gerald.Teschl@univie.ac.at
- Received by editor(s): April 22, 2015
- Received by editor(s) in revised form: August 17, 2015
- Published electronically: December 7, 2015
- Additional Notes: This research was supported by the Austrian Science Fund (FWF) under Grants No. Y330 and W1245
- Communicated by: Joachim Krieger
- © Copyright 2015 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 2 (2015), 51-59
- MSC (2010): Primary 34L25, 35Q41; Secondary 81U30, 81Q15
- DOI: https://doi.org/10.1090/bproc/19
- MathSciNet review: 3450570