Geodesic
Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators
S. A. Denisov1
1 University of Wisconsin–Madison, Mathematics Department 480 Lincoln Dr., Madison, WI, 53706, USA
Mathematical modelling of natural phenomena, Tome 5 (2010) no. 4, pp. 122-149.

Voir la notice de l'article provenant de la source EDP Sciences

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.
DOI : 10.1051/mmnp/20105405

S. A. Denisov 1

1 University of Wisconsin–Madison, Mathematics Department 480 Lincoln Dr., Madison, WI, 53706, USA 
@article{MMNP_2010_5_4_a5,
     author = {S. A. Denisov},
     title = {Wave {Equation} with {Slowly} {Decaying} {Potential:} asymptotics of {Solution} and {Wave} {Operators}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {122--149},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {2010},
     doi = {10.1051/mmnp/20105405},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20105405/}
}
TY  - JOUR
AU  - S. A. Denisov
TI  - Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators
JO  - Mathematical modelling of natural phenomena
PY  - 2010
SP  - 122
EP  - 149
VL  - 5
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20105405/
DO  - 10.1051/mmnp/20105405
LA  - en
ID  - MMNP_2010_5_4_a5
ER  - 
%0 Journal Article
%A S. A. Denisov
%T Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators
%J Mathematical modelling of natural phenomena
%D 2010
%P 122-149
%V 5
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20105405/
%R 10.1051/mmnp/20105405
%G en
%F MMNP_2010_5_4_a5
S. A. Denisov. Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators. Mathematical modelling of natural phenomena, Tome 5 (2010) no. 4, pp. 122-149. doi : 10.1051/mmnp/20105405. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20105405/

[1] M. Christ, A. Kiselev Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials Geom. Funct. Anal. 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. 2006 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. 2004 529 534

[5] S. Denisov, S. Kupin Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight J. Approx. Theory 2006 8 28

[6] S. Denisov Absolutely continuous spectrum of multidimensional Schrödinger operator Int. Math. Res. Not. 2004 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 with L2 potentials Ann. of Math. 2009 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. 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. 1997 203 208

Cité par Sources :