On recent progress of single-realization recoveries of random Schrödinger systems

Shiqi Ma; Shiqi Ma

doi:10.3934/era.2020121

Electronic Research Archive

2021, Volume 29, Issue 3: 2391-2415. doi: 10.3934/era.2020121

Previous Article Next Article

Special Issues

On recent progress of single-realization recoveries of random Schrödinger systems

Shiqi Ma

Department of Mathematics and Statistics, University of Jyväskylä, Finland

Received: 01 May 2020 Revised: 01 October 2020 Published: 24 November 2020
31B10, 35R30, 35S05, 81U40

We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.
- Inverse scattering,
- random source and medium,
- ergodicity,
- pseudodifferential operators,
- microlocal analysis
Citation: Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems[J]. Electronic Research Archive, 2021, 29(3): 2391-2415. doi: 10.3934/era.2020121

Related Papers:

Abstract

We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.

References

[1]	S. Alinhac and P. Gérard, Pseudo-Differential Operators and the Nash-Moser Theorem, volume 82 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2007. Translated from the 1991 French original by Stephen S. Wilson. doi: 10.1090/gsm/082
[2]	Inverse random source scattering problems in several dimensions. SIAM/ASA J. Uncertain. Quantif. (2016) 4: 1263-1287.
[3]	A multi-frequency inverse source problem. J. Differential Equations (2010) 249: 3443-3465.
[4]	Nonradiating sources and transmission eigenfunctions vanish at corners and edges. SIAM J. Math. Anal. (2018) 50: 6255-6270.
[5]	E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, arXiv: 1808.01425, 2018.
[6]	Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. (2002) 111: 230-248.
[7]	Adaptive interferometric imaging in clutter and optimal illumination. Inverse Prob. (2006) 22: 1405-1436.
[8]	Inverse scattering for a random potential. Anal. Appl. (2019) 17: 513-567.
[9]	The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. (2007/08) 30: 1-23.
[10]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences, Springer, Cham, [2019] ⓒ2019. Fourth edition of [MR1183732]. doi: 10.1007/978-3-030-30351-8
[11]	On identifying magnetized anomalies using geomagnetic monitoring. Arch. Ration. Mech. Anal. (2019) 231: 153-187.
[12]	On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model. Arch. Ration. Mech. Anal. (2020) 235: 691-721.
[13]	G. Eskin, Lectures on Linear Partial Differential Equations, volume 123. American Mathematical Society, 2011. doi: 10.1090/gsm/123
[14]	(2016) Introduction to Quantum Mechanics. Cambridge: Cambridge Univ. Press.
[15]	L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Springer, Berlin, second edition edition, 1990. doi: 10.1007/978-3-642-61497-2
[16]	L. Hörmander, The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators, Springer, Berlin, 1994 edition edition, 2007. doi: 10.1007/978-3-540-49938-1
[17]	C. Knox and A. Moradifam, Determining both the source of a wave and its speed in a medium from boundary measurements, Inverse Prob., 36 (2020), 025002, 15 pp. doi: 10.1088/1361-6420/ab53fc
[18]	M. Lassas, L. Päivärinta and E. Saksman, Inverse problem for a random potential, In Partial Differential Equations and Inverse Problems, volume 362 of Contemp. Math., pages 277–288. Amer. Math. Soc., Providence, RI, 2004. doi: 10.1090/conm/362/06618
[19]	Inverse scattering problem for a two dimensional random potential. Comm. Math. Phys. (2008) 279: 669-703.
[20]	Inverse random source problems for time-harmonic acoustic and elastic waves. Comm. Partial Differential Equations (2020) 45: 1335-1380.
[21]	Determining a random Schrödinger equation with unknown source and potential. SIAM J. Math. Anal. (2019) 51: 3465-3491.
[22]	J. Li, H. Liu and S. Ma, Determining a random Schrödinger operator: both potential and source are random, Comm. Math. Phys., 2020. doi: 10.1007/s00220-020-03889-9
[23]	P. Li and X. Wang, An inverse random source problem for maxwell's equations, arXiv: 2002.08732, 2020.
[24]	P. Li and X. Wang, Inverse random source scattering for the Helmholtz equation with attenuation, arXiv: 1911.11189, 2019.
[25]	H. Liu and S. Ma, Single-realization recovery of a random Schrödinger equation with unknown source and potential, arXiv: 2005.04984, 2020.
[26]	H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo-and photo-acoustic tomography, Inverse Prob., 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005
[27]	Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns. Comm. Pure Appl. Math. (2015) 68: 948-963.
[28]	S. Ma, Determination of random Schrödinger operators, Open Access Theses and Dissertations, 671, Hong Kong Baptist University, 2019.
[29]	(2000) Strongly Elliptic Systems and Boundary Integral Equations.Cambridgen University Press.
[30]	C. Pozrikidis, The Fractional Laplacian, Chapman & Hall/CRC, New York, 2016. doi: 10.1201/b19666
[31]	X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Prob., 33 (2017), 035001, 18 pp. doi: 10.1088/1361-6420/aa573c
[32]	M. W. Wong, An Introduction to Pseudo-Differential Operators, volume 6 of Series on Analysis, Applications and Computation, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, third edition, 2014. doi: 10.1142/9074
[33]	G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Prob., 31 (2015), 085003, 13 pp. doi: 10.1088/0266-5611/31/8/085003
[34]	D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Prob., 31 (2015), 035007, 30 pp. doi: 10.1088/0266-5611/31/3/035007

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)