Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials

We prove a family of dispersive estimates for the higher order Schrödinger equation iut=(-Δ)mu+Vu in n spatial dimensions, for m∈N with m>1 and 2m<n<4m. Here V is a real-valued potential belonging to the closure of C0 functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that e-itHPac(H) satisfies a |t|-n2m bound mapping L1 to L∞ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator (-Δ)m+V and a family of dispersive estimates for operators of the form |H|β-n2me-itHPac(H) for 0<β≤n2. The results apply in both even and odd dimensions in the allowed range.