The nonlinear Fourier transform for two-dimensional subcritical potentials

The Novikov-Veselov Equation:Theory and Computation

“…has a nontrivial solution are called exceptional points. It is known that the exceptional points form a bounded closed set in C. Nachman [59] proved that the exceptional set is empty for potentials of conductivity type; more recently, Music [55] has shown that the same is true for subcritical potentials. We discuss this further in section 5.…”

“…19) and (2.20) for smooth potentials of conductivity type is given in section 3 below. If q is smooth, rapidly decreasing, and either critical or subcritical, the Schrödinger potential q can be recovered using the ∂-method of Beals and Coifman [5] (see [45] for the critical case, and [55] for the subcritical case). Both of these papers use techniques developed by Nachman [59] in the context of the inverse conductivity problem.…”

“…The approach taken here is more robust because it works directly with the scattering transform for NV and avoids the Miura-type map. An extension of these ideas to subcritical potentials will appear in [55] and [56].…”

“…According to [55], subcritical potentials do not have nonzero exceptional points. Thus there are no singularities in Figure 11 for parameter values λ ≥ 0.…”

“…al. [9], Tsai [88], Nachman [59], Lassas-Mueller-Siltanen [45], Lassas-Mueller-Siltanen-Stahel [46,47], Music [55], Music-Perry [56], and Perry [67].…”