Asymptotic behaviour of the scattering phase for non-trapping obstacles

Petkov, Veselin; Popov, Georgi

Annales de l'Institut Fourier, Tome 32 (1982) no. 3, pp. 111-149.

Soit $S\left(\lambda \right)$ la matrice de diffusion, associée à l’équation des ondes dans l’extérieur d’un obstacle non-captif $𝒪\subset {\mathbf{R}}^n$, $n\ge 3$ avec condition de Dirichlet ou Neumann sur $\partial 𝒪$. La fonction $s\left(\lambda \right)$, dite phase de diffusion, est déterminée par l’égalité $e^{-2\pi is\left(\lambda \right)}=\det S\left(\lambda \right)$. On démontre que $s\left(\lambda \right)$ admet un développement asymptotique $s\left(\lambda \right)\sim {\sum }_{j=0}^{\infty }{c}_j{\lambda }^{n-j}$ et on calcule les trois premiers coefficients. Notre résultat prouve la conjecture de Majda et Ralston pour des obstacles non-captifs.

Let $S\left(\lambda \right)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪\subset {\mathbf{R}}^n$, $n\ge 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$. The function $s\left(\lambda \right)$, called scattering phase, is determined from the equality $e^{-2\pi is\left(\lambda \right)}=\det S\left(\lambda \right)$. We show that $s\left(\lambda \right)$ has an asymptotic expansion $s\left(\lambda \right)\sim {\sum }_{j=0}^{\infty }{c}_j{\lambda }^{n-j}$ as $\lambda \to +\infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

[1] K. Andersson and R. Melrose, The propagation of singularities along gliding rays, Invent, Math., 41 (1977), 197-232. | MR | Zbl

[2] C. Bardos, J.C. Guillot et J. Ralston, La relation de Poisson pour l'équation des ondes dans un ouvert non-borné. Application à la théorie de la diffusion, preprint. | Numdam | Zbl

[3] C. Bardos, J.C. Guillot et J. Ralston, Relation de Poisson pour l'équation des ondes dans un ouvert non-borné, Séminaire Goulaouic-Schwartz, 1979-1980, exposé n° 13. | Numdam | Zbl

[4] M.S. Birman, Perturbation of the spectrum of a singular elliptic operator under the variation of boundaries and boundary conditions, Dokl. Akad. Nauk SSSR, 137 (1961), 761-763 (in Russian); Soviet Math. Dokl., 2 (1961), 326-328. | Zbl

[5] M.S. Birman, Perturbation of the continuous spectrum of a singular elliptic operator for changing boundary and boundary conditions, Vestnik Leningrad Univ., 1 (1962), 22-55 (in Russian).

[6] M. S. Birman and M.G. Krein, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR, 144 (1962), 475-478 (in Russian). | MR | Zbl

[7] V.S. Buslaev, Scattering plane waves, spectral asymptotics and trace formulas in exterior problems, Dokl. Akad. Nauk SSSR, 197 (1971), 999-1002 (in Russian). | MR | Zbl

[8] Y. Colin De Verdiere, Une formule de trace pour l'opérateur de Schrödinger dans R3, Ann. Scient. Ec. Norm. Sup., 14 (1981), 27-39. | Numdam | MR | Zbl

[9] R. Courant, Uber die Eigenwerte bei den Differentialgleichungen der mathematishen Physik, Math. Z., 7 (1920), 1-57. | JFM

[10] P. Deift, Classical scattering theory with a trace condition, Dissertation, Princeton University, 1976.

[11] I. Gohberg and M.G. Krein, Introduction to the Theory of Linear Non-selfadjoint Operators, AMS Translations, vol 18, Providence 1969. | MR | Zbl

[12] L. Guillope, Une formule de trace pour l'opérateur de Schrödinger dans Rn, thèse, Université Scient. et Medicale Grenoble, 1981.

[13] V. Ia. Ivrii, On the second term of the spectral asymptotics for the Laplace-Beltrami operator on manifold with boundary, Functional Anal. i Pril., 14, n° 2 (1980), 25-34 (in Russian). | MR | Zbl

[14] V.Ia. Ivrii, Sharp spectral asymptotics for the Laplace-Beltrami operator under general ellipic boundary conditions, Functional Anal i Pril., 15, n° 1 (1981), 74-75 (in Russian). | Zbl

[15] A. Jensen and T. Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. in P.D.E., 3 (1978), 1165-1195. | MR | Zbl

[16] T. Kato, Monotonicy theorems in scattering theory, Hadronic Journal, 1 (1978), 134-154. | MR | Zbl

[17] M.G. Krein, On the trace formula in the theory of perturbation, Mat. Sb., 33 (75) (1953), 597-626 (in Russian).

[18] M.G. Krein, On perturbation determinants and a trace formula for unitary and selfadjoint operators, Dokl. Akad. Nauk SSSR, 144 (1962), 268-271. | MR | Zbl

[19] P. Lax and R. Phillips, Scattering theory, Academic Press, 1967. | Zbl

[20] P. Lax and R. Phillips, Scattering theory for the wave equation in even space dimensions, Indiana Univ. Math. J., 22 (1972), 101-134. | MR | Zbl

[21] P. Lax and R. Phillips, The time delay operator and a related trace formula, Topics in Functional Analysis, edited by Gohberg and M. Kac, Academic Press, 1978, p. 197-215. | MR | Zbl

[22] A. Majda, A representation formula for the scattering operator and the inverse problem for arbitrary bodies, Comm. Pure Appl. Math., 30 (1977), 165-194. | MR | Zbl

[23] A. Majda and M. Taylor, Inverse scattering problems for transparant obstacles, electromagnetic waves and hyperbolic systems, Comm. in P.D.E., 2 (1977), 395-433. | MR | Zbl

[24] A. Majda and J. Ralston, An analogue of Weyl's theorem for unbounded domains, I, II, III, Duke Math. J., 45 (1978), 183-196, 513-536 & 46, (1979), 725-731. | Zbl

[25] H.P. Mc Kean and I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geometry, 1, (1967), 43-69. | MR | Zbl

[26] R. Melrose and J. Sjöstrand, Singularities of boundary value problems I, Comm. Pure Appl. Math., 31, (1978), 593-617. | MR | Zbl

[27] R. Melrose and J. Sjöstrand, Singularities of boundary value problems II, Comm. Pure Appl. Math., 35 (1982), 129-168. | MR | Zbl

[28] R. Melrose et J. Sjöstrand, Propagation de singularités pour des problèmes aux limites d'ordre 2, Séminaire Goulaouic-Schwartz, 1977-1978, exposé 15. | Numdam | Zbl

[29] R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J., 46 (1979), 43-59. | MR | Zbl

[30] R. Melrose, Forward scattering by a convex obstacle, Comm. Pure Appl. Math., 33 (1980), 461-499. | MR | Zbl

[31] V. Petkov et G. Popov, Asymptotique de la phase de diffusion pour des domaines non-convexes, C.R. Acad. Sc., Paris, 292 (1981), 275-277. | MR | Zbl

[32] V. Petkov, Comportement asymptotique de la phase de diffusion pour des obstacles non-convexes, Séminaire Goulaouic-Meyer-Schwartz, 1980-1981, Exposé 13. | Numdam | Zbl

[33] Pham The Lai, Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien, Math. Scand., 48 (1981), 5-38. | MR | Zbl

[34] J. Ralston, Diffraction by convex bodies, Séminaire Goulaouic-Schwartz, 1978-1979, Exposé 23. | Numdam | Zbl

[35] J. Ralston, Propagation of singularities and the scattering matrix, In Singularities in boundary value problems, edited by H.G. Garnir, D. Reidel Publ. Company, 1981, p. 169-184. | MR | Zbl

[36] J. Ralston, Note on the decay of acoustic waves, Duke Math. J., 46 (1979), 799-804. | MR | Zbl

[37] M. Reed and B. Simon, Scattering theory, Academic Press, 1979. | Zbl

[38] M. Reed and B. Simon, Analysis of operators, Academic Press, 1978. | Zbl

[39] R. Seeley, A sharp asymptotic remainder estimate for the eigen-values of the laplacian in a domain in R3, Adv. in Math., 29 (1978), 244-269. | MR | Zbl

[40] B. Vainberg, On the short wave asymptotic behavior as t → ∞ of solutions of nonstationary problems, Uspehi Mat. Nauk, 30, n° 2 (1975), 1-55 (in Russian) ; Russian Math. Surveys, 30 (1975), 1-58. | Zbl

[41] H. Weyl, Uber die Asymptotische Verteilung der Eigenwerte, Göttinger Nachr., (1911), 110-117. | JFM