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Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
The nonlinear Fourier transform for two-dimensional subcritical potentials
1. | Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, United States |
References:
[1] |
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasi-Conformal Mappings in the Plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009. |
[2] |
R. Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the $\overline\partial$-method, Inverse Problems, 5 (1989), 87-130.
doi: 10.1088/0266-5611/5/2/002. |
[3] |
M. Boiti, J. Leon, M. Manna and F. Pempinelli, On a spectral transform of a Korteweg-de Vries equation in two spatial dimensions, Inverse Problems, 2 (1986), 271-279.
doi: 10.1088/0266-5611/3/1/008. |
[4] |
R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Communications in Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[5] |
R. Croke, J. Mueller, M. Music, P. Perry, S. Siltanen and A. Stahel, The Novikov-Veselov Equation: Theory and Computation,, , ().
|
[6] |
L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 11 (1966), 209-211. |
[7] |
P. G. Grinevich and R. G. Novikov, Faddeev eigenfunctions for point potentials in two dimensions, Phys. Lett. A, 376 (2012), 1102-1106.
doi: 10.1016/j.physleta.2012.02.025. |
[8] |
M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Communications in Partial Differential Equations, 32 (2007), 591-610.
doi: 10.1080/03605300500530412. |
[9] |
M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I: Analysis, Physica D: Nonlinear Phenomena, 241 (2012), 1322-1335.
doi: 10.1016/j.physd.2012.04.010. |
[10] |
M. Murata, Structure of positive solutions to $(-\Delta+V)u=0$ in $R^n$, Duke Math. J., 53 (1986), 869-943.
doi: 10.1215/S0012-7094-86-05347-0. |
[11] |
M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials, Inverse Problems, 29 (2013), 045004, 25pp.
doi: 10.1088/0266-5611/29/4/045004. |
[12] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96; University of Rochester, Dept. of Mathematics Preprint Series, 19, 1993.
doi: 10.2307/2118653. |
[13] |
P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation, Analysis and Partial Differential Equations, 7 (2014), 311-343.
doi: 10.2140/apde.2014.7.311. |
[14] |
S. Siltanen, Electrical impedance tomography and Faddeev's Green functions, Ann. Acad. Sci. Fenn. Mathematica Dissertationes, 121, (1999), 56pp. |
[15] |
T.-Y. Tsai, The associated evolution equations of the Schödinger operator in the plane, Inverse Problems, 10 (1994), 1419-1432.
doi: 10.1088/0266-5611/10/6/015. |
[16] |
T.-Y. Tsai, The Schrödinger operator in the plane, Inverse Problems, 9 (1993), 763-787.
doi: 10.1088/0266-5611/9/6/012. |
show all references
References:
[1] |
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasi-Conformal Mappings in the Plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009. |
[2] |
R. Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the $\overline\partial$-method, Inverse Problems, 5 (1989), 87-130.
doi: 10.1088/0266-5611/5/2/002. |
[3] |
M. Boiti, J. Leon, M. Manna and F. Pempinelli, On a spectral transform of a Korteweg-de Vries equation in two spatial dimensions, Inverse Problems, 2 (1986), 271-279.
doi: 10.1088/0266-5611/3/1/008. |
[4] |
R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Communications in Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[5] |
R. Croke, J. Mueller, M. Music, P. Perry, S. Siltanen and A. Stahel, The Novikov-Veselov Equation: Theory and Computation,, , ().
|
[6] |
L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 11 (1966), 209-211. |
[7] |
P. G. Grinevich and R. G. Novikov, Faddeev eigenfunctions for point potentials in two dimensions, Phys. Lett. A, 376 (2012), 1102-1106.
doi: 10.1016/j.physleta.2012.02.025. |
[8] |
M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Communications in Partial Differential Equations, 32 (2007), 591-610.
doi: 10.1080/03605300500530412. |
[9] |
M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I: Analysis, Physica D: Nonlinear Phenomena, 241 (2012), 1322-1335.
doi: 10.1016/j.physd.2012.04.010. |
[10] |
M. Murata, Structure of positive solutions to $(-\Delta+V)u=0$ in $R^n$, Duke Math. J., 53 (1986), 869-943.
doi: 10.1215/S0012-7094-86-05347-0. |
[11] |
M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials, Inverse Problems, 29 (2013), 045004, 25pp.
doi: 10.1088/0266-5611/29/4/045004. |
[12] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96; University of Rochester, Dept. of Mathematics Preprint Series, 19, 1993.
doi: 10.2307/2118653. |
[13] |
P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation, Analysis and Partial Differential Equations, 7 (2014), 311-343.
doi: 10.2140/apde.2014.7.311. |
[14] |
S. Siltanen, Electrical impedance tomography and Faddeev's Green functions, Ann. Acad. Sci. Fenn. Mathematica Dissertationes, 121, (1999), 56pp. |
[15] |
T.-Y. Tsai, The associated evolution equations of the Schödinger operator in the plane, Inverse Problems, 10 (1994), 1419-1432.
doi: 10.1088/0266-5611/10/6/015. |
[16] |
T.-Y. Tsai, The Schrödinger operator in the plane, Inverse Problems, 9 (1993), 763-787.
doi: 10.1088/0266-5611/9/6/012. |
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