On the decay properties of solutions to a class of Schrödinger equations
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- by L. Dawson, H. McGahagan and G. Ponce
- Proc. Amer. Math. Soc. 136 (2008), 2081-2090
- DOI: https://doi.org/10.1090/S0002-9939-08-09355-6
- Published electronically: February 14, 2008
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Abstract:
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrödinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections $P_\pm$ onto the positive and negative frequencies.References
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- Walter Craig, Thomas Kappeler, and Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769–860. MR 1361016, DOI 10.1002/cpa.3160480802
- Shin-ichi Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318 (2000), no. 2, 355–389. MR 1795567, DOI 10.1007/s002080000128
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations 31 (2006), no. 10-12, 1811–1823. MR 2273975, DOI 10.1080/03605300500530446
- Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On unique continuation for nonlinear Schrödinger equations, Comm. Pure Appl. Math. 56 (2003), no. 9, 1247–1262. MR 1980854, DOI 10.1002/cpa.10094
- Sigeru Mizohata, On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering, vol. 3, Academic Press, Inc., Orlando, FL; Science Press Beijing, Beijing, 1985. MR 860041
- Luc Molinet and Francis Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with arbitrary large initial data, Int. Math. Res. Not. 70 (2004), 3757–3795. MR 2101982, DOI 10.1155/S107379280414083X
Bibliographic Information
- L. Dawson
- Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721-0089
- Email: ldawson@math.arizona.edu
- H. McGahagan
- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- Email: helena@math.ucsb.edu
- G. Ponce
- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 204988
- Email: ponce@math.ucsb.edu
- Received by editor(s): March 6, 2007
- Published electronically: February 14, 2008
- Additional Notes: The first author was supported by NSF grants
The second author was supported by an NSF postdoctoral fellowship
The third author was supported by NSF grants - Communicated by: Matthew J. Gursky
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2081-2090
- MSC (2000): Primary 35J10; Secondary 35B65
- DOI: https://doi.org/10.1090/S0002-9939-08-09355-6
- MathSciNet review: 2383514