Direct and inverse source scattering for the polyharmonic Schrödinger equation

Yue Zhao

doi:10.3934/dcds.2024156

Article Contents

2025, Volume 45, Issue 7: 2045-2065. Doi: 10.3934/dcds.2024156

This issue Previous Article Next Article Mixing and ergodicity of compositions of inner functions

Direct and inverse source scattering for the polyharmonic Schrödinger equation

Yue Zhao^,

School of Mathematics and Statistics, and Key Lab NAA–MOE, Central China Normal University, Wuhan 430079, China

^*Corresponding author: Yue Zhao
^*Corresponding author: Yue Zhao

Received: August 2024

Revised: September 2024

Early access: November 25, 2024

Published online: November 25, 2024

The author is supported by [NSFC (No. 12371421)].

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

This paper is concerned with the direct and inverse source scattering problems of the polyharmonic Schrödinger equation. For the direct problem, we investigate the meromorphic continuation of the resolvent of the polyharmonic Schrödinger operator $ R_V(\lambda) = ((-\Delta)^{m} + V - \lambda^{2m})^{-1} $ with either bounded or unbounded potentials. A resonance-free region for the resolvent as well as the resolvent estimates in this region are derived for both cases. As an application of the resolvent analysis, an increasing stability is achieved for the inverse source problem of the polyharmonic Schrödinger equation by using the boundary measurements at multiple wavenumbers. The stability estimate contains a Lipschitz type data discrepancy and a high wavenumber tail of the source function. Moreover, the latter decreases as the upper bound of the wavenumber increases.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 78A46.

Citation:

Full Text(HTML)

Figure 1. The region $ \Omega_\delta $ with $ m = 3 $

Download: Full-size image PowerPoint slide

Figure 2. The region $ \mathcal{R} $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	T. Aktosun and V. Papanicolaou, Time evolution of the scattering data for a fourth-order linear differential operator, Inverse Problems, 24 (2008), 055013. doi: 10.1088/0266-5611/24/5/055013.
[2]	G. Bao, J. Lin and F. Triki, Inverse scattering problems withmulti-frequencies, J. Differential Equations, 249 (2010), 3443-3465.
[3]	G. Bao, J. Lin and F. Triki, Inverse scattering problems withmulti-frequencies, Inverse Problems, 31 (2015), 093001, 21 pp. doi: 10.1088/0266-5611/31/9/093001.
[4]	G. Bao, P. Li and Y. Zhao, Stability for the inverse source problems in elasticand electromagnetic waves, J. Math. Pures Appl., 134 (2020), 122-178. doi: 10.1016/j.matpur.2019.06.006.
[5]	N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201. doi: 10.1063/1.523256.
[6]	J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804. doi: 10.1016/j.jde.2015.11.030.
[7]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998.
[8]	A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag., 30 (1982), 1034-1037. doi: 10.1109/TAP.1982.1142902.
[9]	S. Dyatlov and M. Zworski, Mathematical Theory of Scattering Resonances, vol. 200, AMS, 2019.
[10]	H. Feng, A. Soffer, Z. Wu and X. Yao, Decay estimates for higher-order elliptic operators, Trans. Amer. Math. Soc., 373 (2020), 2805-2859. doi: 10.1090/tran/8010.
[11]	A. Fokas, Y. Kurylev and V. Marinakis, The unique determination of neuronal currents in the brain via magnetoencephalography, Inverse Problems, 20 (2004), 1067-1082. doi: 10.1088/0266-5611/20/4/005.
[12]	F. Gazzola, H.-C. Grunau and G. Sweers, Boundary Value Problems, Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2010.
[13]	S. Huang, X. Yao and Q. Zheng, Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems, Inverse Problems, 30 (2018), 43-55. doi: 10.1515/forum-2016-0162.
[14]	K. Iwasaki, Scattering theory for 4th order differential operators: Ⅰ-Ⅱ, Japan. J. Math., 14 (1988), 1-96. doi: 10.4099/math1924.14.59.
[15]	R. Lavine and A. Nachman, unpublished, announced in A. I. Nachman, Inverse scattering at fixed energy, Mathematical Physics, X, 14 (1991), 434-441. doi: 10.1007/978-3-642-77303-7_48.
[16]	P. Li, X. Yao and Y. Zhao, Stability for an inverse source problem of the biharmonic operator, SIAM J. Appl. Math., 81 (2021), 2503-2525. doi: 10.1137/21M1407148.
[17]	P. Li, X. Yao and Y. Zhao, The scattering resonances for Schrödinger-type operators with unbounded potentials, SIAM J. Math. Anal., 56 (2024), 2149-2170. doi: 10.1137/22M1498619.
[18]	P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Problems and Imaging, 11 (2017), 745-759. doi: 10.3934/ipi.2017035.
[19]	P. Li, J. Zhai and Y. Zhao, Stability for the acoustic inverse source problem in inhomogeneous media, SIAM J. Appl. Math., 80 (2020), 2547-2559. doi: 10.1137/20M1334267.
[20]	H. Mizutani and X. Yao, Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials, arXiv: 2004.10115v1, (2020).
[21]	H. Mizutani and X. Yao, Kato smoothing and Strichartz estimates for higher-order and fractional operators with sharp Hardy potentials, Comm. Math. Phys., 388 (2021), 581-623. doi: 10.1007/s00220-021-04229-1.
[22]	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.
[23]	E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, LPrinceton University Press, Princeton, New Jersey, 1993.
[24]	T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227. doi: 10.3934/ipi.2018008.
[25]	T. Tyni, M. Harju and V. Serov, Recovery of singularities in a fourth-order operator on the line from limited data, Inverse Problems, 32 (2016), 075001, 22 pp. doi: 10.1088/0266-5611/32/7/075001.
[26]	H. Weyl, Das asymptotische verteilungsgesetz der eigenwerte linear partieller differentialgleichungen, Math. Ann., 71 (1911), 441-479. doi: 10.1007/BF01456804.