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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

1. 

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

Received  January 2008 Revised  June 2008 Published  July 2008

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.
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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