Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and C2 potential

Journal of Differential Equations

Volumn 228, Issue 2, 2006, Pages 440-485

  Castella, François ^a   Jecko, Thierry ^a  

^a UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Besov space; Helmholtz equation; Limiting absorption principle; Morrey Campanato spaces; Non trapping; Wigner transform

Indexed keywords

EID: 33746538453     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.06.006     Document Type: Article

References (32)

1. Agmon S. Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 4 (1975) 151-218
2. Agmon S., and Hörmander L. Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math. 30 (1976) 1-37
3. 0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians (1996), Birkhäuser
4. Benamou J.D., Castella F., Katsaounis T., and Perthame B. High frequency limit in the Helmholtz equation. Rev. Mat. Iberoamericana 18 1 (2002) 187-209 and Séminaire E.D.P., École Polytechnique, exposé N. V, 1999/2000, 27 pp
5. Burq N. Semi-classical estimates for the resolvent in non-trapping geometries. Int. Math. Res. Not. 5 (2002) 221-241
6. Castella F. The radiation condition at infinity for the high-frequency Helmholtz equation with source term: A wave packet approach. J. Funct. Anal. 223 (2005) 204-257
7. Castella F., Perthame B., and Runborg O. High frequency limit of the Helmholtz equation II: Source on a general smooth manifold. Comm. Partial Differential Equations 27 3/4 (2002) 607-651
8. F. Castella, O. Runborg, in preparation
9. Cycon H.L., Froese R.G., Kirsch W., and Simon B. Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry (1987), Springer
10. De Bièvre S., and Pravica D.W. Spectral analysis for optical fibres and stratified fluids I. The limiting absorption principle. J. Funct. Anal. 98 2 (1991) 404-436
11. Dereziński J., and Gérard C. Scattering Theory of Classical and Quantum N-Particle Systems. Texts Monogr. Phys. (1997), Springer, Berlin
12. Dimassi M., and Sjöstrand J. Spectral Asymptotics in the Semiclassical Limit. London Math. Soc. Lecture Note Ser. vol. 268 (1999), Cambridge Univ. Press, Cambridge
13. E. Fouassier, Thesis, University Rennes 1, 2005
14. Gérard P., and Leichtnam E. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 2 (1993) 559-607
15. Gérard P. Microlocal defect measures. Comm. Partial Differential Equations 16 11 (1991) 1761-1794
16. Gérard C., and Martinez A. Principe d'absorption limite pour des opérateurs de Schrödinger à longue portée. C. R. Acad. Sci. Paris Sér. I Math. 306 3 (1988) 121-123
17. Jecko Th. Estimations de la résolvante pour une molécule diatomique dans l'approximation de Born-Oppenheimer. Comm. Math. Phys. 195 3 (1998) 585-612
18. Jecko T. From classical to semi-classical non-trapping behaviour: A new proof. C. R. Acad. Sci. Paris Ser. I Math. 338 (2004) 545-548
19. Jecko T. Non-trapping condition for semiclassical Schrödinger operators with matrix-valued potentials. Math. Phys. Electron. J. 11 2 (2005)
20. Jensen A., Mourre E., and Perry P. Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. I.H.P. 41 2 (1984) 207-225
21. Klein M., Martinez A., and Wang X.P. On the Born-Oppenheimer approximation of wave operators in molecular scattering. Comm. Math. Phys. 152 (1993) 73-95
22. Klein M., Martinez A., and Wang X.P. On the Born-Oppenheimer approximation of diatomic wave operators II. Singular potentials. J. Math. Phys. 38 3 (1997)
23. Lions P.L., and Paul Th. Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 3 (1993) 553-618
24. Lions P.L., and Perthame B. Lemmes de moments, de moyenne, et de dispersion. C. R. Acad. Sci. Paris Sér. I Math. 314 11 (1992) 801-806
25. Martinez A. An Introduction to Semicassical and Microlocal Analysis. Universitext (2002), Springer, New York
26. Morawetz C.S. Time decay for the non-linear Klein-Gordon equation. Proc. R. Soc. Lond. Ser. A 306 (1968) 291-296
27. Mourre E. Absence of singular continuous spectrum for certain self-adjoint operators. Comm. Math. Phys. 78 (1981) 391-408
28. Perthame B., and Vega L. Morrey-Campanato estimates for Helmholtz equation. J. Funct. Anal. 164 2 (1999) 340-355
29. B. Perthame, L. Vega, Sommerfeld radiation condition for Helmholtz equation with variable index at infinity, preprint
30. Robert D., and Tamura H. Semiclassical estimates for resolvents and asymptotics for total cross-section. Ann. IHP 46 (1987) 415-442
31. Wang X.P. Time decay of scattering solutions and resolvent estimates for semiclassical Schrödinger operators. J. Differential Equations 71 (1988) 348-395
32. Wang X.P., and Zhang P. High frequency limit of the Helmholtz equation with variable refraction index. J. Funct. Anal. 230 1 (2006) 116-168