This paper is devoted to the study of the limit measures associated with the solution of the Helmholtz equation with a source term which concentrates on a point. The potential is assumed to be and the operator non trapping. The solution of the semi-classical Schrödinger equation is written micro-locally as a finite sum of Lagrangian distributions. Under a geometrical hypothesis, which generalizes the virial assumption, this representation implies that the limit measure exists and satisfies standard properties. Finally, one gives an example of operator which does not satisfy the geometrical hypothesis and for which the limit measure is not unique. The case of two source terms is also treated
Cet article est consacré à l’étude des mesures limites associées à la solution de l’équation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est supposé et l’opérateur non-captif. La solution de l’équation de Schrödinger semi-classique s’écrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothèse géométrique, qui généralise l’hypothèse du viriel, on en déduit que la mesure limite existe et qu’elle vérifie des propriétés standard. Enfin, on donne un exemple d’opérateur qui ne vérifie pas l’hypothèse géométrique et pour lequel la mesure limite n’est pas unique. Le cas de deux termes sources est aussi traité.
@article{AFST_2009_6_18_3_445_0,
author = {Jean-Fran\c{c}ois Bony},
title = {Mesures limites pour l{\textquoteright}\'equation de {Helmholtz} dans le cas non captif},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {445--479},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {6e s{\'e}rie, 18},
number = {3},
year = {2009},
doi = {10.5802/afst.1210},
mrnumber = {2582438},
zbl = {1186.35032},
language = {fr},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1210/}
}
TY - JOUR AU - Jean-François Bony TI - Mesures limites pour l’équation de Helmholtz dans le cas non captif JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2009 SP - 445 EP - 479 VL - 18 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1210/ DO - 10.5802/afst.1210 LA - fr ID - AFST_2009_6_18_3_445_0 ER -
%0 Journal Article %A Jean-François Bony %T Mesures limites pour l’équation de Helmholtz dans le cas non captif %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2009 %P 445-479 %V 18 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1210/ %R 10.5802/afst.1210 %G fr %F AFST_2009_6_18_3_445_0
Jean-François Bony. Mesures limites pour l’équation de Helmholtz dans le cas non captif. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 3, pp. 445-479. doi : 10.5802/afst.1210. https://afst.centre-mersenne.org/articles/10.5802/afst.1210/
[1] Benamou (J. D.), Castella (F.), Katsaounis (T.), Perthame (B.).— High frequency limit of the Helmholtz equations, Rev. Mat. Iberoamericana 18, no 1, p. 187–209 (2002). | MR | Zbl
[2] Burq (N.).— Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not., no 5, p. 221–241 (2002). | MR | Zbl
[3] Castella (F.).— The radiation condition at infinity for the high-frequency Helmholtz equation with source term : a wave-packet approach, J. Funct. Anal. 223, no 1, p. 204–257 (2005). | MR | Zbl
[4] Castella (F.), Jecko (T.).— Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and potential, J. Differential Equations 228, no 2, p. 440–485 (2006). | MR | Zbl
[5] Castella (F.), Jecko (T.), Knauf (A.).— Semiclassical resolvent estimates for Schrödinger operators with coulomb singularities, Ann. Henri Poincaré 9, n 4, p. 775–815 (2008). | MR | Zbl
[6] Castella (F.), Perthame (B.), Runborg (O.).— High frequency limit of the Helmholtz equation. II. Source on a general smooth manifold, Comm. Partial Differential Equations 27, no 3-4, p. 607–651 (2002). | MR
[7] Dimassi (M.), Sjöstrand (J.).— Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press (1999). | MR | Zbl
[8] Fedoriuk (M.), Maslov (V.).— Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics, vol. 7, D. Reidel Publishing Co., Dordrecht (1981). | MR | Zbl
[9] Fouassier (E.).— High frequency analysis of Helmholtz equations : case of two point sources, SIAM J. Math. Anal. 38, no 2, p. 617–636 (2006). | MR | Zbl
[10] Fouassier (E.).— High frequency limit of Helmholtz equations : refraction by sharp interfaces, J. Math. Pures Appl. (9) 87, no 2, p. 144–192 (2007). | MR | Zbl
[11] Gérard (C.).— A proof of the abstract limiting absorption principle by energy estimates, J. Funct. Anal. 254, n 11, p. 2707–2724 (2008). | MR | Zbl
[12] Gérard (C.), Martinez (A.).— Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris Sér. I Math. 306, no 3, p. 121–123 (1988). | MR | Zbl
[13] Gérard (C.), Martinez (A.).— Semiclassical asymptotics for the spectral function of long-range Schrödinger operators, J. Funct. Anal. 84, no 1, p. 226–254 (1989). | MR | Zbl
[14] Hörmander (L.).— The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 256, Springer-Verlag, Distribution theory and Fourier analysis (1990). | MR | Zbl
[15] Hörmander (L.).— The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften, vol. 275, Springer-Verlag, Fourier integral operators, Corrected reprint of the 1985 original (1994). | MR | Zbl
[16] Isozaki (H.), Kitada (H.).— Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32, no 1, p. 77–104 (1985). | MR | Zbl
[17] Isozaki (H.), Kitada (H.).— A remark on the microlocal resolvent estimates for two body Schrödinger operators, Publ. Res. Inst. Math. Sci. 21, no 5, p. 889–910 (1985). | MR | Zbl
[18] Ivrii (V.).— Microlocal analysis and precise spectral asymptotics, Springer Monographs in Mathematics, Springer-Verlag (1998). | MR | Zbl
[19] Mourre (E.).— Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys. 78, no 3, p. 391–408 (1980/81). | MR | Zbl
[20] Perthame (B.), Vega (L.).— Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal. 164, no 2, p. 340–355 (1999). | MR | Zbl
[21] Robert (D.).— Autour de l’approximation semi-classique, Progress in Mathematics, vol. 68, Birkhäuser Boston Inc. (1987). | MR | Zbl
[22] Robert (D.), Tamura (H.).— Semiclassical estimates for resolvents and asymptotics for total scattering cross-sections, Ann. Inst. H. Poincaré Phys. Théor. 46, no 4, p. 415–442 (1987). | Numdam | MR | Zbl
[23] Robert (D.), Tamura (H.).— Semi-classical asymptotics for local spectral densities and time delay problems in scattering processes, J. Funct. Anal. 80, no 1, p. 124–147 (1988). | MR | Zbl
[24] Robert (D.), Tamura (H.).— Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits, Ann. Inst. Fourier 39, no 1, p. 155–192 (1989). | Numdam | MR | Zbl
[25] Wang (X. P.).— Microlocal estimates of the Schrödinger equation in semi-classical limit, Partial differential equations and applications, Sémin. congr., vol. 15 Math. France, p. 265–308, (2007). | MR | Zbl
[26] Wang (X. P.), Zhang (P.).— High-frequency limit of the Helmholtz equation with variable refraction index, J. Funct. Anal. 230, no 1, p.116–168 (2006). | MR | Zbl
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