I would like to know if there is any known result on dispersive estimates for Schrodinger operator with magnetic potential in one dimension. There is a lot of literature for three dimensional magnetic operators, but i was not able to find anything in dimension one.
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$\begingroup$ what is a "magnetic potential" ? do you mean a vector potential? $\endgroup$– Carlo BeenakkerJan 25, 2017 at 20:31
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$\begingroup$ By Schrodinger operator with magnetic potential i mean an operator of the form $(i\nabla-A)^2$ $\endgroup$– CapublancaJan 30, 2017 at 2:39
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In 1D, a change of variables is sufficient to transform a first order perturbation (i.e., a magnetic potential) into a 0th order perturbation (i.e. an electric potential). Thus you can apply the results concerning the electric case; there are several. Actually this trick works even for the coefficient of the second order derivative, so you can reduce the fully variable coefficients case to the electric potential case. See Proposition 1.1 in this paper where a dispersive estimate in presence of a magnetic potential is proved. Let me add that this result is probably not sharp (I need one derivative decaying like x^{-2}), and by a more direct method it might be possible to relax the smoothness of the coefficients.
answered Jan 26, 2017 at 7:24