Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy. (English) Zbl 1424.35288

Green, William R.; Toprak, Ebru

Differ. Integral Equ. 30, No. 5-6, 329-386 (2017).

Summary: We investigate dispersive estimates for the Schrödinger operator \(H=-\Delta +V\) 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 \[ e^{itH}\chi(H)P_{ac}(H)=O(\tfrac{1}{\log t})A_0+O(\tfrac{1}{t})A_1+O((t\log t)^{-1})A_2+O(t^{-1}(\log t)^{-2})A_3. \] Here, \(A_0,A_1\colon L^1(\mathbb{R}^4)\to L^\infty (\mathbb{R}^4)\), while \(A_2\), \(A_3\) are operators between logarithmically weighted spaces, with \(A_0\), \(A_1\), \(A_2\) 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 eigenvalue one can recover the \(|t|^{-2}\) bound as an operator from \(L^1\to L^\infty\). Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

MSC:

35Q41	Time-dependent Schrödinger equations and Dirac equations
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords:

canonical resonance function; finite rank operator; time decay; resolvent expansion; global dispersive bound