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

We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ 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 $\|F_t\|_{L^1\to L^\infty} \lesssim 1/\log t$ for $t>2$ such that $\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for} t>2.$ 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.

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