2023/06/15 by Makoto Ikoma, Ikoma, Makoto, Soichiro SUZUKI +1
Mathematics · #Advanced Mathematical Physics Problems #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2306.08982
Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schrödinger equations and Dirac equations. For d-dimensional Schrödinger-type equations (d ≥ 2), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for d-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schrödinger-type equations. Indeed, Ikoma obtained the optimal constant in the case d = 2, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for d-dimensional Schrödinger-type and Dirac equations with radial initial data for any d ≥ 2. In addition, we also give optimal constants for the one-dimensional Schrödinger-type and Dirac equations.