Time-Decay Estimates for Two-Dimensional Fourth-Order Schrödinger Operators with Threshold Singularities

We establish time-decay estimates for the two-dimensional fourth-order Schrödinger operator $H={\mathrm{\Delta }}^2+V$ 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 $${\parallel H^{\frac{\alpha }{4}}{e}^{-itH}{P}_{\mathrm{a}\mathrm{c}}(H)\parallel }_{L^1\to L^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{2+\alpha }{4}},-2<\alpha \le 2,$$ which matches with the free sharp decay rate throughout the full range of $\alpha$. For a second-kind resonance, the decay rate is $\mathrm{\mid }t{\mathrm{\mid }}^{-(2+\alpha )\mathrm{/}4}(\log (2+\mathrm{\mid }t\mathrm{\mid }){)}^2$ for every $-2<\alpha \le 2$, 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 $(\log \mathrm{\mid }t\mathrm{\mid }{)}^{-1}$ for $\alpha =0$ and $\mathrm{\mid }t{\mathrm{\mid }}^{-\alpha \mathrm{/}4}(\log \mathrm{\mid }t\mathrm{\mid }{)}^{-2}$ for $0<\alpha \le 2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<\alpha \le 2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<\alpha \le 2$ and $s>0$: $${\parallel {\omega }^{-s}{H}^{\frac{\alpha }{4}}{e}^{-itH}{P}_{\mathrm{a}\mathrm{c}}(H){\omega }^{-s}\parallel }_{L^1\to L^{\mathrm{\infty }}}\lesssim \frac{1}{\mathrm{\mid }t{\mathrm{\mid }}^{\frac{2+\alpha }{4}}(\log \mathrm{\mid }t\mathrm{\mid }{)}^s},\mathrm{\mid }t\mathrm{\mid }\ge 2,$$ where $\omega (x)=\log (2+\mathrm{\mid }x\mathrm{\mid })$. By contrast, zero is a second-kind resonance for the free operator ${\mathrm{\Delta }}^2$, 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.