Resonances and balls in obstacle scattering with Neumann boundary conditions

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August 2008, 2(3): 335-340. doi: 10.3934/ipi.2008.2.335

Resonances and balls in obstacle scattering with Neumann boundary conditions

T. J. Christiansen ^1,

1.	Department of Mathematics, University of Missouri, Columbia, Missouri 65211, United States

Received January 2008 Revised June 2008 Published July 2008

Abstract
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We consider scattering by a smooth obstacle in $R^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.

Keywords: resonances, inverse problems..

Mathematics Subject Classification: Primary: 35P25; Secondary: 47A40, 81U4.

Citation: T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems and Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335

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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