-
Previous Article
Parametrices for the light ray transform on Minkowski spacetime
- IPI Home
- This Issue
-
Next Article
On the parameter estimation problem of magnetic resonance advection imaging
Scattering problems for perturbations of the multidimensional biharmonic operator
Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland |
Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space $H_{-δ}^2 $. The main result of this paper is the proof of Saito's formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito's formula is based on norm estimates for the resolvent of the direct operator in $H_{-δ}^1 $.
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann.Scuola Norm.Sup.Pisa, 2 (1975), 151-218.
|
[2] |
T. Aktosun and V. G. Papanicolaou, Time-evolution of the scattering data for a fourth-order linear differential operator,
Inverse Problems 24 (2008), 055013, 14 pp. |
[3] |
Y. Assylbekov, Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order,
Inverse Problems 32 (2016), 105009, 22 pp. |
[4] |
Y. Assylbekov, Corrigendum: Inverse problems for the perturbed polyharmonic operator with coefficients in {S}obolev spaces with non-positive order,
Inverse Problems 33 (2017), 099501. |
[5] |
J. Bergh and J. Löfström,
Interpolation Spaces: An Introduction Springer-Verlag, New York, 1976. |
[6] |
F. Cakoni and D. Colton,
A Qualitative Approach to Inverse Scattering Theory Springer, New York, 2014. |
[7] |
G. Eskin,
Lectures on Linear Partial Differential Equations American Mathematical Society, 2011. |
[8] |
L. Evans,
Partial Differential Equations American Mathematical Society, 2010. |
[9] |
F. Gazzola, H. -C. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems Springer-Verlag Berlin Heidelberg, 2010. |
[10] |
M. Harju,
On the Direct and Inverse Scattering Problems for a Nonlinear Three-dimensional Schrödinger Equation Ph. D thesis, University of Oulu, 2010. |
[11] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators: Differential Operators with Constant Coefficients Springer-Verlag Berlin Heidelberg, 2005. |
[12] |
K. Iwasaki,
Scattering theory for the 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.
doi: 10.4099/math1924.14.1. |
[13] |
K. Iwasaki,
Scattering theory for the 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.
doi: 10.4099/math1924.14.1. |
[14] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans.Amer.Math.Soc., 366 (2014), 95-112.
|
[15] |
S. T. Kuroda,
Finite-dimensional perturbation and a representation of scattering operator, Pacific J. Math., 13 (1963), 1305-1318.
doi: 10.2140/pjm.1963.13.1305. |
[16] |
N. N. Lebedev,
Special Functions and Their Applications Dover Publications, Inc., New York, 1972. |
[17] |
N. V. Movchan, R. C. McPhedran, A. B. Movchan and C. G. Poulton,
Wave scattering by platonic grating stacks, Proc.R.Soc. A., 465 (2009), 3383-3400.
doi: 10.1098/rspa.2009.0301. |
[18] |
L. Päivärinta and V. Serov,
Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711.
doi: 10.1137/S0036141096305796. |
[19] |
L. Päivärinta and V. Serov,
New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential, Inverse Problems, 17 (2001), 1321-1326.
doi: 10.1088/0266-5611/17/5/306. |
[20] |
B. Pausader,
Scattering for the defocusing beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791-822.
doi: 10.1512/iumj.2010.59.3966. |
[21] |
Y. Saito,
Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math., 19 (1982), 527-547.
|
[22] |
V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator,
Inverse Problems 32 (2016), 045002, 19pp. |
[23] |
V. Serov and M. Harju,
A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 1323-1337.
doi: 10.1088/0951-7715/21/6/010. |
[24] |
Z. Sun,
An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc, 338 (1993), 953-969.
|
[25] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[26] |
G. Watson,
A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. |
show all references
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann.Scuola Norm.Sup.Pisa, 2 (1975), 151-218.
|
[2] |
T. Aktosun and V. G. Papanicolaou, Time-evolution of the scattering data for a fourth-order linear differential operator,
Inverse Problems 24 (2008), 055013, 14 pp. |
[3] |
Y. Assylbekov, Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order,
Inverse Problems 32 (2016), 105009, 22 pp. |
[4] |
Y. Assylbekov, Corrigendum: Inverse problems for the perturbed polyharmonic operator with coefficients in {S}obolev spaces with non-positive order,
Inverse Problems 33 (2017), 099501. |
[5] |
J. Bergh and J. Löfström,
Interpolation Spaces: An Introduction Springer-Verlag, New York, 1976. |
[6] |
F. Cakoni and D. Colton,
A Qualitative Approach to Inverse Scattering Theory Springer, New York, 2014. |
[7] |
G. Eskin,
Lectures on Linear Partial Differential Equations American Mathematical Society, 2011. |
[8] |
L. Evans,
Partial Differential Equations American Mathematical Society, 2010. |
[9] |
F. Gazzola, H. -C. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems Springer-Verlag Berlin Heidelberg, 2010. |
[10] |
M. Harju,
On the Direct and Inverse Scattering Problems for a Nonlinear Three-dimensional Schrödinger Equation Ph. D thesis, University of Oulu, 2010. |
[11] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators: Differential Operators with Constant Coefficients Springer-Verlag Berlin Heidelberg, 2005. |
[12] |
K. Iwasaki,
Scattering theory for the 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.
doi: 10.4099/math1924.14.1. |
[13] |
K. Iwasaki,
Scattering theory for the 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.
doi: 10.4099/math1924.14.1. |
[14] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans.Amer.Math.Soc., 366 (2014), 95-112.
|
[15] |
S. T. Kuroda,
Finite-dimensional perturbation and a representation of scattering operator, Pacific J. Math., 13 (1963), 1305-1318.
doi: 10.2140/pjm.1963.13.1305. |
[16] |
N. N. Lebedev,
Special Functions and Their Applications Dover Publications, Inc., New York, 1972. |
[17] |
N. V. Movchan, R. C. McPhedran, A. B. Movchan and C. G. Poulton,
Wave scattering by platonic grating stacks, Proc.R.Soc. A., 465 (2009), 3383-3400.
doi: 10.1098/rspa.2009.0301. |
[18] |
L. Päivärinta and V. Serov,
Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711.
doi: 10.1137/S0036141096305796. |
[19] |
L. Päivärinta and V. Serov,
New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential, Inverse Problems, 17 (2001), 1321-1326.
doi: 10.1088/0266-5611/17/5/306. |
[20] |
B. Pausader,
Scattering for the defocusing beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791-822.
doi: 10.1512/iumj.2010.59.3966. |
[21] |
Y. Saito,
Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math., 19 (1982), 527-547.
|
[22] |
V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator,
Inverse Problems 32 (2016), 045002, 19pp. |
[23] |
V. Serov and M. Harju,
A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 1323-1337.
doi: 10.1088/0951-7715/21/6/010. |
[24] |
Z. Sun,
An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc, 338 (1993), 953-969.
|
[25] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[26] |
G. Watson,
A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. |
[1] |
Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 |
[2] |
Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064 |
[3] |
Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026 |
[4] |
Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 |
[5] |
Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 |
[6] |
Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291 |
[7] |
Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029 |
[8] |
Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455 |
[9] |
Lili Yan. Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022034 |
[10] |
Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211 |
[11] |
Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems and Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951 |
[12] |
Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681 |
[13] |
Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035 |
[14] |
Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 |
[15] |
John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 |
[16] |
Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062 |
[17] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 |
[18] |
Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 |
[19] |
Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems and Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004 |
[20] |
Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]