Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators

S. A. Denisov

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 122-149
  • ISSN: 0973-5348

Abstract

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In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.

How to cite

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Denisov, S. A.. "Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators." Mathematical Modelling of Natural Phenomena 5.4 (2010): 122-149. <http://eudml.org/doc/197704>.

@article{Denisov2010,
abstract = {In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.},
author = {Denisov, S. A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {wave equation; square-summable potential; wave operators; one space dimension; long-time asymptotics; spectral representation},
language = {eng},
month = {5},
number = {4},
pages = {122-149},
publisher = {EDP Sciences},
title = {Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators},
url = {http://eudml.org/doc/197704},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Denisov, S. A.
TI - Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 122
EP - 149
AB - In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.
LA - eng
KW - wave equation; square-summable potential; wave operators; one space dimension; long-time asymptotics; spectral representation
UR - http://eudml.org/doc/197704
ER -

References

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  1. M. Christ, A. Kiselev. Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials. Geom. Funct. Anal., 12 (2002), 1174–1234. 
  2. D. Damanik, B. Simon. Jost functions and Jost solutions for Jacobi matrices. I. A necessary and sufficient condition for Szegő asymptotics. Invent. Math., 165 (2006), No. 1, 1–50. 
  3. S. Denisov. On weak asymptotics for Schrödinger evolution. Mathematical Modelling of Natural Phenomena (to appear).  
  4. S. Denisov. On the existence of wave operators for some Dirac operators with square summable potential. Geom. Funct. Anal., 14 (2004), No. 3, 529–534. 
  5. S. Denisov, S. Kupin. Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight. J. Approx. Theory, 139 (2006), No. 1–2, 8–28. 
  6. S. Denisov. Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not., 2004, No. 74, 3963–3982.  
  7. R. Killip. Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not., 2002, 2029–2061.  
  8. R. Killip, B. Simon. Sum rules and spectral measure of Schrödinger operators withL2 potentials. Ann. of Math., (2) 170 (2009), No. 2, 739–782.  
  9. P. Lax, R. Phillips. Scattering theory. Pure and Applied Mathematics, Academic Press Inc., Boston, 1989.  
  10. S.N. Naboko. Dense point spectra of Schrödinger and Dirac operators. Theor. Mat. Fiz., 68 (1986), 18–28. 
  11. B. Simon. Orthogonal polynomials on the unit circle. Parts 1 and 2. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 2005.  
  12. B. Simon. Some Schrödinger operators with dense point spectrum. Proc. Amer. Math. Soc., 125 (1997), 203–208. 

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