Dispersion estimates for 2D Dirac equation (Q2856522)

The paper deals with the two-dimensional linear Dirac equation with the Maxwell potential and aims at establishing the rate of the long-time decay of the solution of the Cauchy problem. The potential in the equation is assumed to satisfy a certain spatial decay condition and the ``regular case'' condition saying that there is neither resonance no eigenvalue at the thresholds. The main result is that under these conditions, the norm of the solution in the associated weighted Sobolev spaces decays as \(| t|^{-1}\log ^{-2}| t|\), \(t\to\pm\infty\). It is also shown that the solutions of the free Dirac equation (with zero potential), which violated the ``regular case'' condition, decay only as \(t^{-1}\). The idea of the proof is to consider separately the low energy and high energy components of the solution. Then the high energy component shows a fast decay, as \(t^{-N}\) with any \(N>0\) whereas for the low energy component, a suitable modification of the method by Jensen and Kato is proposed, which is based on new asymptotics for the perturbed resolvent and a 2D version of the Jensen-Kato-Zygmund lemma on ``one-and-half partial integration''.