Decay Estimates for Four Dimensional Schrödinger, Klein-gordon and Wave Equations with Obstructions at Zero Energy
نویسندگان
چکیده
We investigate dispersive estimates for the Schrödinger operator H = −∆+V with V is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion eχ(H)Pac(H) = O(1/(log t))A0 +O(1/t)A1 +O((t log t) )A2 +O(t (log t))A3. Here A0, A1 : L (R) → L∞(Rn), while A2, A3 are operators between logarithmically weighted spaces, with A0, A1, A2 finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigevalue one can recover the |t|−2 bound as an operator from L → L∞. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.
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