Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy

Abstract

We investigate $L^1({\mathbb{R}}^4)\to L^{\infty }({\mathbb{R}}^4)$ dispersive estimates for the Schrödinger operator $H=-\Delta +V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\Vert F_t{\Vert }_{L^1\to L^{\infty }}\lesssim 1/\log t$ for $t>2$ such that $$\Vert e^{itH}{P}_{ac}-F_t{\Vert }_{L^1\to L^{\infty }}\lesssim t^{-1},\text{for}t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.

32 pages

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Cited by in corpus (14)

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• On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances
• Attractors of Hamilton nonlinear partial differential equations
• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions
• A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy
• Time decay estimates for the wave equation with potential in dimension two
• On the $L^p$ boundedness of wave operators for four-dimensional Schrödinger Operators with a threshold eigenvalue
• Abstract theory of decay estimates: perturbed Hamiltonians
• Dispersive Estimates in R^3 with Threshold Resonances
• On Absence of Threshold Resonances for Schrodinger and Dirac Operators
• Intertwining wave operators, Fourier restriction, and Wiener theorems
• The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates