Time-Decay Estimates for Two-Dimensional Fourth-Order Schr\"odinger Operators with Threshold Singularities

Abstract

We establish time-decay estimates for the two-dimensional fourth-order Schr\"odinger operator $H=\Delta^2+V$ with a real-valued decaying potential $V$, covering all possible zero-energy threshold obstructions. When zero is a regular point or a first-kind resonance, we prove \[ \left\| H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H) \right\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{2+\alpha}{4}}, \qquad -2<\alpha\leq2, \] which matches with the free sharp decay rate throughout the full range of $\alpha$. For a second-kind resonance, the decay rate is $|t|^{-(2+\alpha)/4}(\log(2+|t|))^2$ for every $-2<\alpha\leq2$, with only a logarithmic loss. For the stronger threshold singularities, we show that the large-time behavior is governed by the presence of a \(d\)-wave resonance. If zero is a third-kind resonance, or an eigenvalue accompanied by a \(d\)-wave resonance, we obtain the sharp decay $(\log|t|)^{-1}$ for $\alpha=0$ and $|t|^{-\alpha/4}(\log|t|)^{-2}$ for $0<\alpha\leq2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<\alpha\leq2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<\alpha\leq2$ and $s>0$: \[ \left\| \omega^{-s} H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H)\omega^{-s} \right\|_{L^1\to L^\infty} \lesssim \frac{1} {|t|^{\frac{2+\alpha}{4}}(\log|t|)^s}, \qquad |t|\geq2, \] where $\omega(x)=\log(2+|x|)$. By contrast, zero is a second-kind resonance for the free operator $\Delta^2$, and the free evolution admits no such logarithmic gain. Thus, in the regular and first-kind cases, the potential changes the zero-energy spectral structure of the free operator, and this change is accompanied by improved weighted decay.