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Lp−Lp Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential

✍ Scribed by Ricardo Weder

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
286 KB
Volume
170
Category
Article
ISSN
0022-1236

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✦ Synopsis

In this paper I prove a L p &L p estimate for the solutions to the one-dimensional Schro dinger equation with a potential in L 1 # where in the generic case #>3Â2 and in the exceptional case (i.e., when there is a half-bound state of zero energy) #>5Â2. I use this estimate to construct the scattering operator for the nonlinear Schro dinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity.

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