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November  2014, 8(4): 1151-1167. doi: 10.3934/ipi.2014.8.1151

## The nonlinear Fourier transform for two-dimensional subcritical potentials

 1 Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, United States

Received  November 2013 Revised  June 2014 Published  November 2014

The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax the conductivity type assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scattering transform can be inverted, and the original potential is recovered from the scattering data.
Citation: Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems and Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151
##### 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, arXiv:1312.5427, to appear in Contemporary Mathematics. [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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##### 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, arXiv:1312.5427, to appear in Contemporary Mathematics. [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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