August  2021, 29(3): 2391-2415. doi: 10.3934/era.2020121

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

Department of Mathematics and Statistics, University of Jyväskylä, Finland

Received  May 2020 Revised  October 2020 Published  August 2021 Early access  November 2020

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.

Citation: Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, 2021, 29 (3) : 2391-2415. doi: 10.3934/era.2020121
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.  Google Scholar

[2]

G. BaoC. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 1263-1287.  doi: 10.1137/16M1067470.  Google Scholar

[3]

G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

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E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.  Google Scholar

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E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, arXiv: 1808.01425, 2018. Google Scholar

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P. BlomgrenG. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.  doi: 10.1121/1.1421342.  Google Scholar

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L. BorceaG. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Prob., 22 (2006), 1405-1436.  doi: 10.1088/0266-5611/22/4/016.  Google Scholar

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P. CaroT. Helin and M. Lassas, Inverse scattering for a random potential, Anal. Appl., 17 (2019), 513-567.  doi: 10.1142/S0219530519500015.  Google Scholar

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C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2007/08), 1-23.  doi: 10.1137/06066970X.  Google Scholar

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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.  Google Scholar

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Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

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G. Eskin, Lectures on Linear Partial Differential Equations, volume 123. American Mathematical Society, 2011. doi: 10.1090/gsm/123.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. LassasL. Päivärinta and E. Saksman, Inverse scattering problem for a two dimensional random potential, Comm. Math. Phys., 279 (2008), 669-703.  doi: 10.1007/s00220-008-0416-6.  Google Scholar

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J. LiT. Helin and P. Li, Inverse random source problems for time-harmonic acoustic and elastic waves, Comm. Partial Differential Equations, 45 (2020), 1335-1380.  doi: 10.1080/03605302.2020.1774895.  Google Scholar

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J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[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.  Google Scholar

[23]

P. Li and X. Wang, An inverse random source problem for maxwell's equations, arXiv: 2002.08732, 2020. Google Scholar

[24]

P. Li and X. Wang, Inverse random source scattering for the Helmholtz equation with attenuation, arXiv: 1911.11189, 2019. Google Scholar

[25]

H. Liu and S. Ma, Single-realization recovery of a random Schrödinger equation with unknown source and potential, arXiv: 2005.04984, 2020. Google Scholar

[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.  Google Scholar

[27]

Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math., 68 (2015), 948-963.  doi: 10.1002/cpa.21503.  Google Scholar

[28]

S. Ma, Determination of random Schrödinger operators, Open Access Theses and Dissertations, 671, Hong Kong Baptist University, 2019. Google Scholar

[29] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridgen University Press, 2000.   Google Scholar
[30]

C. Pozrikidis, The Fractional Laplacian, Chapman & Hall/CRC, New York, 2016. doi: 10.1201/b19666.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

G. BaoC. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 1263-1287.  doi: 10.1137/16M1067470.  Google Scholar

[3]

G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

[4]

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.  Google Scholar

[5]

E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, arXiv: 1808.01425, 2018. Google Scholar

[6]

P. BlomgrenG. Papanicolaou and H. Zhao, Super-resolution in time-reversal acoustics, J. Acoust. Soc. Am., 111 (2002), 230-248.  doi: 10.1121/1.1421342.  Google Scholar

[7]

L. BorceaG. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Prob., 22 (2006), 1405-1436.  doi: 10.1088/0266-5611/22/4/016.  Google Scholar

[8]

P. CaroT. Helin and M. Lassas, Inverse scattering for a random potential, Anal. Appl., 17 (2019), 513-567.  doi: 10.1142/S0219530519500015.  Google Scholar

[9]

C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2007/08), 1-23.  doi: 10.1137/06066970X.  Google Scholar

[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.  Google Scholar

[11]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

[12]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

[13]

G. Eskin, Lectures on Linear Partial Differential Equations, volume 123. American Mathematical Society, 2011. doi: 10.1090/gsm/123.  Google Scholar

[14] D. J. Griffiths, Introduction to Quantum Mechanics, Cambridge Univ. Press, Cambridge, 2016.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. LassasL. Päivärinta and E. Saksman, Inverse scattering problem for a two dimensional random potential, Comm. Math. Phys., 279 (2008), 669-703.  doi: 10.1007/s00220-008-0416-6.  Google Scholar

[20]

J. LiT. Helin and P. Li, Inverse random source problems for time-harmonic acoustic and elastic waves, Comm. Partial Differential Equations, 45 (2020), 1335-1380.  doi: 10.1080/03605302.2020.1774895.  Google Scholar

[21]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[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.  Google Scholar

[23]

P. Li and X. Wang, An inverse random source problem for maxwell's equations, arXiv: 2002.08732, 2020. Google Scholar

[24]

P. Li and X. Wang, Inverse random source scattering for the Helmholtz equation with attenuation, arXiv: 1911.11189, 2019. Google Scholar

[25]

H. Liu and S. Ma, Single-realization recovery of a random Schrödinger equation with unknown source and potential, arXiv: 2005.04984, 2020. Google Scholar

[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.  Google Scholar

[27]

Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math., 68 (2015), 948-963.  doi: 10.1002/cpa.21503.  Google Scholar

[28]

S. Ma, Determination of random Schrödinger operators, Open Access Theses and Dissertations, 671, Hong Kong Baptist University, 2019. Google Scholar

[29] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridgen University Press, 2000.   Google Scholar
[30]

C. Pozrikidis, The Fractional Laplacian, Chapman & Hall/CRC, New York, 2016. doi: 10.1201/b19666.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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