Local smoothing estimates for the massless Dirac–Coulomb equation in 2 and 3 dimensions

Abstract

Abstract. We prove local smoothing estimates for the massless Dirac equation with a Coulomb potential in 2 and 3 dimensions. Our strategy is inspired by [9] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.

Dispersive estimates for the Dirac equation in an Aharonov–Bohm field

“…Proof of Theorem 1.1. Our proof is a combination of the arguments used in [9] with the ones in [8], and follows the strategy originally developed in [5], the idea being use decomposition (2.1) to reduce equation (1.3) to a much simpler problem, use Propositions (2.3) and (2.4) to prove the local smoothing estimate for a fixed value of l ∈ Z and then sum back. We thus set an initial condition f ∈ L 2 with angular part in h l and denote with L l f the solution to the Cauchy problem…”

“…We stress the fact that the magnetic potential A is critical with respect to the scaling of the massless Dirac operator; as it is well known, the study of dispersive estimates for flows perturbed by scaling-critical potentials represents a particularly interesting and challenging problem, as perturbation arguments typically do not work in this setting. In this framework we mention [5]- [6] in which smoothing and Strichartz estimates for the Schrödinger and wave equations with inverse square potential are proved, then [14] in which the L 1 → L ∞ time decay is proved for a wide class of Schrödinger flows with critical electromagnetic potentials, later [9] in which local smoothing estimates for the Dirac-Coulomb equation is discussed, and finally [8] which is devoted to the study of weak dispersion for fractional Aharonov-Bohm-Schrödinger groups. On the other hand, the study of dispersive estimates for the Dirac equation perturbed with small magnetic potential has been developed e.g.…”

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We prove local smoothing and weighted Strichartz estimates for the Dirac equation with a Aharonov-Bohm potential. The proof, inspired by [9], relies on an explicit representation of the solution built in terms of spectral projections.

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A Dirac field interacting with point nuclear dynamics

“…We stress the fact that, in contrast with the Schrödinger case, a severe difficulty here is represented by the strong singularity produced by the moving nuclei: indeed, the Coulomb potential exhibits the same homogeneity of the (massless) Dirac operator or, in other words, it is critical with respect to the natural scaling of the operator. This is the source of several problems, especially from the point of view of dispersive dynamics: indeed, it is not known whether Strichartz estimates hold for the flow e it(D+meβ+ ν |x| ) , even in the case m e = 0 (we mention the papers [10] in which a family of local smoothing estimates for such a flow is proved and [9] in which the same result is obtained in the case of Aharonov-Bohm fields), while it is interesting to notice that for the scaling critical non-relativistic counterpart, i.e. the Schrödinger equation with inverse square potential, the dispersive dynamics is now completely understood (see [6] for Strichartz, and [19] for time-decay estimates in even more general settings).…”

Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential

“…We refer to [19,36,1,11,7,8] and references therein for further references on such weakdispersive estimates for dispersive equations with singular perturbations. More recently, Suzuki [46] proved Strichartz estimates for all admissible pairs except for the endpoint (p, q) = (2, 2n/(n− 2)) and used them to study the well-posedness of nonlinear Schrödinger equations.…”