American Institute of Mathematical Sciences

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

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.   Google Scholar

[2]

A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'', Classics of Soviet Mathematics, (1996).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Hassell and M. Zworski, Resonant rigidity of s2,, J. Funct. Anal., 169 (1999), 604.  doi: 10.1006/jfan.1999.3487.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. B. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle,, Journées Équations aux Dérivées partielles (1984), (1984), 1.   Google Scholar

[11]

R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures., Cambridge University Press, (1995).   Google Scholar

[12]

V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'', Pure and Applied Mathematics (New York). John Wiley & Sons, ().   Google Scholar

[13]

V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant,, Ann. Henri Poincaré, 2 (2001), 675.  doi: 10.1007/PL00001049.  Google Scholar

[14]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'', Academic Press [Harcourt Brace Jovanovich, (1978).   Google Scholar

[15]

D. Robert, On the Weyl formula for obstacles,, in, (1995), 264.   Google Scholar

[16]

J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729.   Google Scholar

[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.  Google Scholar

[18]

M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).   Google Scholar

[19]

M. Zworski, Poisson formulae for resonances,, Séminaire sur les Équations aux Dérivées Partielles, (1997), 1996.   Google Scholar

[20]

M. Zworski, Poisson formula for resonances in even dimensions,, Asian J. Math., 2 (1998), 609.   Google Scholar

show all references

References:
[1]

A. D. Alexandrov, To the theory of mixed volumes of convex bodies part II,, Mat. Sbornik, 2 (1937), 1205.   Google Scholar

[2]

A. D. Alexandrov, "Selected Works. Part I. Selected Scientific Papers,'', Classics of Soviet Mathematics, (1996).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Hassell and M. Zworski, Resonant rigidity of s2,, J. Funct. Anal., 169 (1999), 604.  doi: 10.1006/jfan.1999.3487.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. B. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle,, Journées Équations aux Dérivées partielles (1984), (1984), 1.   Google Scholar

[11]

R. B. Melrose, "Geometric Scattering Theory,'' Stanford Lectures., Cambridge University Press, (1995).   Google Scholar

[12]

V. Petkov and L. Stoyanov, "Geometry of Reflecting Rays and Inverse Spectral Problems,'', Pure and Applied Mathematics (New York). John Wiley & Sons, ().   Google Scholar

[13]

V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant,, Ann. Henri Poincaré, 2 (2001), 675.  doi: 10.1007/PL00001049.  Google Scholar

[14]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators,'', Academic Press [Harcourt Brace Jovanovich, (1978).   Google Scholar

[15]

D. Robert, On the Weyl formula for obstacles,, in, (1995), 264.   Google Scholar

[16]

J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729.   Google Scholar

[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.  Google Scholar

[18]

M. E. Taylor, "Partial Differential Equations. II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).   Google Scholar

[19]

M. Zworski, Poisson formulae for resonances,, Séminaire sur les Équations aux Dérivées Partielles, (1997), 1996.   Google Scholar

[20]

M. Zworski, Poisson formula for resonances in even dimensions,, Asian J. Math., 2 (1998), 609.   Google Scholar

[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]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[7]

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
[8]

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
[9]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[10]

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
[11]

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
[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]

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
[14]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607
[15]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297
[16]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423
[17]

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
[18]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[19]

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
[20]

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

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences