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

top
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

top

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

top
  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. Zbl1039.34076
  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. Zbl1122.47029
  3. S. Denisov. On weak asymptotics for Schrödinger evolution. Mathematical Modelling of Natural Phenomena (to appear).  Zbl1193.35019
  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. Zbl1070.47504
  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. Zbl1099.41025
  6. S. Denisov. Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not., 2004, No. 74, 3963–3982.  Zbl1077.35104
  7. R. Killip. Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not., 2002, 2029–2061.  Zbl1021.34071
  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.  Zbl1185.34131
  9. P. Lax, R. Phillips. Scattering theory. Pure and Applied Mathematics, Academic Press Inc., Boston, 1989.  Zbl0697.35004
  10. S.N. Naboko. Dense point spectra of Schrödinger and Dirac operators. Theor. Mat. Fiz., 68 (1986), 18–28. Zbl0607.34023
  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. Zbl0888.34071

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.