Resonances and balls in obstacle scattering with Neumann boundary conditions

Previous Article
An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions
IPI Home
This Issue
Next Article
Identifiability and reconstruction of shapes from integral invariants

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

Abstract
Related Papers

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 & 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.
[2]	A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'', Classics of Soviet Mathematics, (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. 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. 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. 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. doi: 10.1016/0001-8708(79)90042-2.
[7]	A. Hassell and M. Zworski, Resonant rigidity of s²,, J. Funct. Anal., 169 (1999), 604. 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. 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. 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), (1984), 1.
[11]	R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures., Cambridge University Press, (1995).
[12]	V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'', Pure and Applied Mathematics (New York). John Wiley & Sons, ().
[13]	V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant,, Ann. Henri Poincaré, 2 (2001), 675. doi: 10.1007/PL00001049.
[14]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'', Academic Press [Harcourt Brace Jovanovich, (1978).
[15]	D. Robert, On the Weyl formula for obstacles,, in, (1995), 264.
[16]	J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729.
[17]	J. Sjöstrand and M. Zworski, Lower bounds on the number of scattering poles, II,, J. Funct. Anal., 123 (1994), 336. doi: 10.1006/jfan.1994.1092.
[18]	M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).
[19]	M. Zworski, Poisson formulae for resonances,, Séminaire sur les Équations aux Dérivées Partielles, (1997), 1996.
[20]	M. Zworski, Poisson formula for resonances in even dimensions,, Asian J. Math., 2 (1998), 609.

show all references

References:

[1]	A. D. Alexandrov, To the theory of mixed volumes of convex bodies part II,, Mat. Sbornik, 2 (1937), 1205.
[2]	A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'', Classics of Soviet Mathematics, (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. 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. 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. 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. doi: 10.1016/0001-8708(79)90042-2.
[7]	A. Hassell and M. Zworski, Resonant rigidity of s²,, J. Funct. Anal., 169 (1999), 604. 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. 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. 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), (1984), 1.
[11]	R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures., Cambridge University Press, (1995).
[12]	V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'', Pure and Applied Mathematics (New York). John Wiley & Sons, ().
[13]	V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant,, Ann. Henri Poincaré, 2 (2001), 675. doi: 10.1007/PL00001049.
[14]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'', Academic Press [Harcourt Brace Jovanovich, (1978).
[15]	D. Robert, On the Weyl formula for obstacles,, in, (1995), 264.
[16]	J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729.
[17]	J. Sjöstrand and M. Zworski, Lower bounds on the number of scattering poles, II,, J. Funct. Anal., 123 (1994), 336. doi: 10.1006/jfan.1994.1092.
[18]	M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).
[19]	M. Zworski, Poisson formulae for resonances,, Séminaire sur les Équations aux Dérivées Partielles, (1997), 1996.
[20]	M. Zworski, Poisson formula for resonances in even dimensions,, Asian J. Math., 2 (1998), 609.

[1]	Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225
[2]	Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139
[3]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
[4]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215
[5]	Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
[6]	Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[7]	Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[8]	Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[9]	Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[10]	Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[11]	Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[12]	Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[13]	Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[14]	François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289
[15]	Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[16]	Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[17]	Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[18]	Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[19]	Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems & Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397
[20]	Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371

2017 Impact Factor: 1.465

American Institute of Mathematical Sciences