2026/07/26 by Jiabin Qian, Manli Song
#math.AP
Let La=-Δx-Δy+(a)/(2)|x|-2 with a>0 denote the Schrödinger operator on L2(ℝ2x× ℝny), which involves a singular partial inverse-square potential. The purpose of this manuscript is twofold. First, relying on the explicit representation for the spectral measure associated with the operator La established by Zhang-Zhang [J. Geom. Anal. 35(3), Paper No. 71, 27pp (2025)], we investigate the decay estimate for a class of dispersive semigroups of the form eitϕ(√(La)), where ϕ: ℝ+ → ℝ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function ϕ, we adopt the frequency localization and the stationary phase method. In the second part of the paper, we first derive boundary Strichartz estimates for the fractional Schrödinger operator eitLaν, 0<ν≠(1)/(2). As applications of the established decay estimates, we further obtain Strichartz estimates for some concrete wave equations associated with the operator La, which corresponds to ϕ(r)=r, r2, r2+r4, √(1+r2), √(1+r4), and rμ,0<μ≠ 1. Most notably, our results unify and simplify existing dispersive estimates for the operator La, while extending the relevant theory to more general scenarios.