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An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions
Resonances and balls in obstacle scattering with Neumann boundary conditions
1. | Department of Mathematics, University of Missouri, Columbia, Missouri 65211, United States |
References:
[1] |
A. D. Alexandrov, To the theory of mixed volumes of convex bodies part II, Mat. Sbornik, 2 (1937), 1205-1238. |
[2] |
A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'' Classics of Soviet Mathematics, 4. Gordon and Breach Publishers, Amsterdam, 1996. |
[3] |
C. Bardos, J.-C. Guillot and J. Ralston, La relation de Poisson pour l'équation des ondes dans un ouvert non borné. Application à la théorie de la diffusion, Comm. Partial Differential Equations, 7 (1982), 905-958.
doi: 10.1080/03605308208820241. |
[4] |
T. Branson and P. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations, 15 (1990), 245-272.
doi: 10.1080/03605309908820686. |
[5] |
T. Christiansen, Spectral asymptotics for compactly supported perturbations of the Laplacian on Rn, Comm. Partial Differential Equations, 23 (1998), 933-948.
doi: 10.1080/03605309808821373. |
[6] |
V. Guillemin and R. B. Melrose, The Poisson summation formula for manifolds with boundary, Adv. in Math., 32 (1979), 204-232.
doi: 10.1016/0001-8708(79)90042-2. |
[7] |
A. Hassell and M. Zworski, Resonant rigidity of s2, J. Funct. Anal., 169 (1999), 604-609.
doi: 10.1006/jfan.1999.3487. |
[8] |
R. B. Melrose, Scattering theory and the trace of the wave group, J. Funct. Anal., 45 (1982), 29-40.
doi: 10.1016/0022-1236(82)90003-9. |
[9] |
R. B. Melrose, Polynomial bound on the number of scattering poles, J. Funct. Anal., 53 (1983), 287-303.
doi: 10.1016/0022-1236(83)90036-8. |
[10] |
R. B. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle, Journées Équations aux Dérivées partielles (1984), 1-8. |
[11] |
R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures. Cambridge University Press, Cambridge, 1995. |
[12] |
V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'' Pure and Applied Mathematics (New York). John Wiley & Sons, Ltd., Chichester. |
[13] |
V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant, Ann. Henri Poincaré, 2 (2001), 675-711.
doi: 10.1007/PL00001049. |
[14] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'' Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. |
[15] |
D. Robert, On the Weyl formula for obstacles, in "Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995),'' 264-285, Progr. Nonlinear Differential Equations Appl., 21, Birkhuser Boston, Boston, MA, 1996. |
[16] |
J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., 4 (1991), 729-769. |
[17] |
J. Sjöstrand and M. Zworski, Lower bounds on the number of scattering poles, II, J. Funct. Anal., 123 (1994), 336-367.
doi: 10.1006/jfan.1994.1092. |
[18] |
M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'' Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. |
[19] |
M. Zworski, Poisson formulae for resonances, Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. XIII, 14pp., École Polytech., Palaiseau, 1997. |
[20] |
M. Zworski, Poisson formula for resonances in even dimensions, Asian J. Math., 2 (1998), 609-617. |
show all references
References:
[1] |
A. D. Alexandrov, To the theory of mixed volumes of convex bodies part II, Mat. Sbornik, 2 (1937), 1205-1238. |
[2] |
A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'' Classics of Soviet Mathematics, 4. Gordon and Breach Publishers, Amsterdam, 1996. |
[3] |
C. Bardos, J.-C. Guillot and J. Ralston, La relation de Poisson pour l'équation des ondes dans un ouvert non borné. Application à la théorie de la diffusion, Comm. Partial Differential Equations, 7 (1982), 905-958.
doi: 10.1080/03605308208820241. |
[4] |
T. Branson and P. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations, 15 (1990), 245-272.
doi: 10.1080/03605309908820686. |
[5] |
T. Christiansen, Spectral asymptotics for compactly supported perturbations of the Laplacian on Rn, Comm. Partial Differential Equations, 23 (1998), 933-948.
doi: 10.1080/03605309808821373. |
[6] |
V. Guillemin and R. B. Melrose, The Poisson summation formula for manifolds with boundary, Adv. in Math., 32 (1979), 204-232.
doi: 10.1016/0001-8708(79)90042-2. |
[7] |
A. Hassell and M. Zworski, Resonant rigidity of s2, J. Funct. Anal., 169 (1999), 604-609.
doi: 10.1006/jfan.1999.3487. |
[8] |
R. B. Melrose, Scattering theory and the trace of the wave group, J. Funct. Anal., 45 (1982), 29-40.
doi: 10.1016/0022-1236(82)90003-9. |
[9] |
R. B. Melrose, Polynomial bound on the number of scattering poles, J. Funct. Anal., 53 (1983), 287-303.
doi: 10.1016/0022-1236(83)90036-8. |
[10] |
R. B. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle, Journées Équations aux Dérivées partielles (1984), 1-8. |
[11] |
R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures. Cambridge University Press, Cambridge, 1995. |
[12] |
V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'' Pure and Applied Mathematics (New York). John Wiley & Sons, Ltd., Chichester. |
[13] |
V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant, Ann. Henri Poincaré, 2 (2001), 675-711.
doi: 10.1007/PL00001049. |
[14] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'' Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. |
[15] |
D. Robert, On the Weyl formula for obstacles, in "Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995),'' 264-285, Progr. Nonlinear Differential Equations Appl., 21, Birkhuser Boston, Boston, MA, 1996. |
[16] |
J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., 4 (1991), 729-769. |
[17] |
J. Sjöstrand and M. Zworski, Lower bounds on the number of scattering poles, II, J. Funct. Anal., 123 (1994), 336-367.
doi: 10.1006/jfan.1994.1092. |
[18] |
M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'' Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. |
[19] |
M. Zworski, Poisson formulae for resonances, Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. XIII, 14pp., École Polytech., Palaiseau, 1997. |
[20] |
M. Zworski, Poisson formula for resonances in even dimensions, Asian J. Math., 2 (1998), 609-617. |
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