Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

We consider the Cauchy problem for the two-dimensional Novikov-Veselov equation integrable via the inverse scattering problem for the Schr\"odinger operator with fixed negative energy. The associated linear equation is characterized by a rational symbol which is not a polynomial, except when the energy parameter is zero. With the help of a complex analysis point of view of the problem, we establish uniform decay estimates for the linear solution with gain of almost one derivative, and we use this result together with Fourier decomposition methods and $X^{s,b}$ spaces to prove local well-posedness in $H^s$, $s>\frac12$.