The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

Castella, François

Journées équations aux dérivées partielles (2004), article no. 4, 18 p.

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\alpha >0$. The high-frequency (or: semi-classical) parameter is $\epsilon >0$. We let $\epsilon$ and $\alpha$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.

Under these assumptions, we prove that the solution $u^{\epsilon }$ radiates in the outgoing direction, uniformly in $\epsilon$. In particular, the function $u^{\epsilon }$, when conveniently rescaled at the scale $\epsilon$ close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform (in $\epsilon$) version of the limiting absorption principle.

Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) For very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in $\epsilon$.

[Ag] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa, Vol. 2, N. 4, pp. 151-218 (1975). | Numdam | MR | Zbl

[AH] S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. Vol. 30, pp. 1-37 (1976). | MR | Zbl

[BCKP] J.D. Benamou, F. Castella, Th. Katsaounis, B. Perthame, High frequency limit in the Helmholtz equation, Rev. Mat. Iberoamericana, Vol. 18, N. 1, pp. 187-209 (2002), and, Séminaire E.D.P., École Polytechnique, exposé N. V, 27 pp., 1999-2000. | MR | Zbl

[BR] A. Bouzouina, D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Vol. 111, no. 2, pp. 223-252 (2002). | MR | Zbl

[Bu] N. Burq, Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not., no. 5, pp. 221-241 (2002). | MR | Zbl

[Bt] J. Butler, Global H Fourier Integral operators with complex valued phase functions, preprint Bologna (2001).

[C] R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. Henri Poincaré, Vol. 3, N. 4, pp 757-772 (2002). | MR | Zbl

[CPR] F. Castella, B. Perthame, O. Runborg, High frequency limit of the Helmholtz equation II: source on a general smooth manifold, Comm. P.D.E., Vol. 27, N. 3-4, pp. 607-651 (2002). | MR | Zbl

[Ca] F. Castella, The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach, preprint.

[CRu] F. Castella, O. Runborg, In preparation.

[CRo] M. Combescure, D. Robert, Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptot. Anal., Vol. 14, no. 4, pp. 377-404 (1997). | MR | Zbl

[CRR] M. Combescure, J. Ralston, D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Comm. Math. Phys., Vol. 202, no. 2, pp. 463-480 (1999). | MR | Zbl

[DG] J. Dereziński, C. Gérard, Scattering theory of classical and quantum $N$-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin (1997). | MR | Zbl

[DS] M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semiclassical Limit, London Math. Soc. Lecture Notes series 268, Cambridge Universuty Press (1999). | MR | Zbl

[Dsf] D. Dos Santos Ferreira, Inégalités de Carleman $L^p$ pour des indices critiques, Thesis, Université de Rennes 1 (2002).

[Fo] G.B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122. Princeton University Press, Princeton, NJ (1989). | MR | Zbl

[Fou] E. Fouassier, Thesis, University Rennes 1, in preparation.

[GM] C. Gérard, A. Martinez, Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris Sér. I Math., Vol. 306, no. 3, pp. 121-123 (1988). | MR | Zbl

[HJ] G.A. Hagedorn, A. Joye, Semiclassical dynamics with exponentially small error estimates, Comm. Math. Phys., Vol. 207, pp. 439-465 (1999). | MR | Zbl

[H1] G.A. Hagedorn, Semiclassical quantum mechanics III, Ann. Phys., Vol. 135, pp. 58-70 (1981). | MR

[H2] G.A. Hagedorn, Semiclassical quantum mechanics IV, Ann. IHP, Vol. 42, pp. 363-374 (1985). | Numdam | MR | Zbl

[HR] B. Helffer, D. Robert, Caclcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal., Vol. 53, N. 3, pp. 246-268 (1983). | MR | Zbl

[He] K. Hepp, The classical limit of quantum mechanical correlation functions, Comm. Math. Phys., Vol. 35, pp. 265-277 (1974). | MR

[Ho] L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. Vol. 219, N. 3, pp. 413-449 (1995). | MR | Zbl

[J] T. Jecko, From classical to semi-classical non-trapping behaviour: a new proof, Preprint University of Rennes 1 (2002).

[Ma] A. Martinez, An introduction to Semicassical and Microlocal Analysis, Universitext, Springer-Verlag, New York (2002). | MR | Zbl

[PV1] B. Perthame, L. Vega, Morrey-Campanato estimates for Helmholtz Equation, J. Funct. Anal., Vol. 164, N. 2, pp. 340-355 (1999). | MR | Zbl

[PV2] B. Perthame, L. Vega, Sommerfeld radiation condition for Helmholtz equation with variable index at infinity, Preprint (2002).

[Ro] D. Robert, Remarks on asymptotic solutions for time dependent Schrödinger equations, to appear. | Zbl

[Ro2] D. Robert, notes on lectures given at the University of Nantes (1999-2000).

[Rb] S. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., Vol. 34, pp. 2150-2205 (1993). | MR | Zbl

[Wa] X.P. Wang, Time decay of scattering solutions and resolvent estimates for Semiclassical Schrödinger operators, J. Diff. Eq., Vol. 71, pp. 348-395 (1988). | MR | Zbl

[WZ] X.P. Wang, P. Zhang, High frequency limit of the Helmholtz equation with variable refraction index, Preprint (2004).