Scattering problems for perturbations of the multidimensional biharmonic operator

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February 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 and 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.
[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.

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