Abstract
We investigate $L^1\\to L^\\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\\frac{1}{2}}$ decay rate, which may be improved to $t^{-\\frac{1}{2} - \\gamma}$ for any $0\\leq \\gamma<\\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.
Highlights
We consider the linear Dirac equation with a potential: (1)i∂tψ(x, t) = (Dm + V (x))ψ(x, t), ψ(x, 0) = ψ0(x).Here the spatial variable x ∈ R2, and ψ(x, t) ∈ C2
We show that the classification of resonances for the massless Dirac equation and their dynamical consequences do not follow the same patterns as the Schrodinger equation
In the case of Dirac equation, as in Schrodinger equation, the time decay can be improved at the cost of spatial weights
Summary
We consider the linear Dirac equation with a potential:. i∂tψ(x, t) = (Dm + V (x))ψ(x, t), ψ(x, 0) = ψ0(x). We note that the assumption of a lack of embedded eigenvalues is not needed for our low energy results in Theorem 1.1, as the spectral properties in a neighborhood of zero are dictated by the threshold behavior. The lack of embedded eigenvalues has been established in the massive case, [8], and in the massless case for a sufficiently small potential, [11].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
