American Institute of Mathematical Sciences

Advanced
January  2018, 12(1): 205-227. doi: 10.3934/ipi.2018008

Scattering problems for perturbations of the multidimensional biharmonic operator

Teemu Tyni , and  Valery Serov

Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

* Corresponding author: Teemu Tyni

Received  February 2017 Revised  August 2017 Published  December 2017

Fund Project: This work was supported by the Academy of Finland (application number 250215, Finnish Programme for Centres of Excellence in Research 2012-2017)

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 $.

Keywords: Saito's formula, biharmonic operator, scattering problem, inverse problem, perturbation.
Mathematics Subject Classification: Primary: 47A40, 34A55; Secondary: 81U40.
Citation: Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008
References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann.Scuola Norm.Sup.Pisa, 2 (1975), 151-218. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[5]

J. Bergh and J. Löfström, Interpolation Spaces: An Introduction Springer-Verlag, New York, 1976. Google Scholar

[6]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Springer, New York, 2014. Google Scholar

[7]

G. Eskin, Lectures on Linear Partial Differential Equations American Mathematical Society, 2011. Google Scholar

[8]

L. Evans, Partial Differential Equations American Mathematical Society, 2010. Google Scholar

[9]

F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems Springer-Verlag Berlin Heidelberg, 2010. Google Scholar

[10]

M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-dimensional Schrödinger Equation Ph. D thesis, University of Oulu, 2010.Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators: Differential Operators with Constant Coefficients Springer-Verlag Berlin Heidelberg, 2005.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

N. N. Lebedev, Special Functions and Their Applications Dover Publications, Inc., New York, 1972. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

Y. Saito, Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math., 19 (1982), 527-547. Google Scholar

[22]

V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems 32 (2016), 045002, 19pp. Google Scholar

[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. Google Scholar

[24]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc, 338 (1993), 953-969. Google Scholar

[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. Google Scholar

[26]

G. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. Google Scholar

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann.Scuola Norm.Sup.Pisa, 2 (1975), 151-218. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[5]

J. Bergh and J. Löfström, Interpolation Spaces: An Introduction Springer-Verlag, New York, 1976. Google Scholar

[6]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Springer, New York, 2014. Google Scholar

[7]

G. Eskin, Lectures on Linear Partial Differential Equations American Mathematical Society, 2011. Google Scholar

[8]

L. Evans, Partial Differential Equations American Mathematical Society, 2010. Google Scholar

[9]

F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems Springer-Verlag Berlin Heidelberg, 2010. Google Scholar

[10]

M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-dimensional Schrödinger Equation Ph. D thesis, University of Oulu, 2010.Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators: Differential Operators with Constant Coefficients Springer-Verlag Berlin Heidelberg, 2005.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

N. N. Lebedev, Special Functions and Their Applications Dover Publications, Inc., New York, 1972. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

Y. Saito, Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math., 19 (1982), 527-547. Google Scholar

[22]

V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems 32 (2016), 045002, 19pp. Google Scholar

[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. Google Scholar

[24]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc, 338 (1993), 953-969. Google Scholar

[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. Google Scholar

[26]

G. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. Google Scholar

[1]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[2]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[3]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[4]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[5]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
[6]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[7]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[8]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[9]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[10]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[11]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[12]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[13]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[14]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[15]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[16]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[17]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[18]

Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems & Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057
[19]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[20]

Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (66)
  • HTML views (319)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences