带记忆的完全分数阶薛定谔方程的延拓理论
An extension theory for fully fractional Schrödinger equations with memory
- Arizona State University（亚利桑那州立大学）
- Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对带记忆的完全分数阶薛定谔方程，通过构建算子半群、等价局部延拓问题及振荡泊松核，结合配套理论建立了非线性记忆问题的适定性理论。
AI中文摘要：
我们通过完全分数阶薛定谔算子$(\u2202_t-i\u0394_x)^s$（其中$0<s<1$）的延拓理论来研究带记忆的非线性薛定谔方程。由于该算子在空间和时间上均是非局部的，其适定的柯西问题需给定过去的历史数据，而非经典的初始条件。我们构建了$(\u2202_t-i\u0394_x)^s$的半群定义，推导了在一个额外空间变量下的等价局部延拓问题，并显式计算了对应的振荡泊松核。该延拓方法可对历史数据进行泊松提升，并在历史数据上建立一个内在二次型，从而在提升的加权体能量与完全由给定过去演化确定的非局部能量之间建立显式恒等式。作为应用，结合我们配套论文中发展的非线性边界相互作用理论，我们得到了相关非线性记忆问题的适定性理论。该适定性在延拓诱导的温和意义下建立：对于自然能量空间中的一般历史数据，未来演化被定义为半空间中对应解的边界迹；在额外的图域假设下，该分数阶方程在$L^2$空间及正时间内可被恢复。
英文摘要：
We develop a Caffarelli-Silvestre extension theory for the fully fractional Schrödinger operator $\mathcal{L}^s=(\partial_t-iΔ_x)^s$, $0<s<1$, nonlocal in space and time, whose Cauchy problem prescribes a past history rather than data at a single time. Every extension theory so far rests on positivity or sectoriality of the generator; here the semigroup is unitary and the symbol changes sign across the characteristic paraboloid, so neither is at hand. We construct the extension nonetheless. Its Poisson kernel, computed explicitly, is oscillatory rather than positive; its Dirichlet-to-Neumann map is $\mathcal{L}^s$; its normalising constant is the Caffarelli-Silvestre constant times the phase $e^{i\frac{πs}{2}}$; and the theory undergoes a transition at $s=\frac12$. We then identify the intrinsic Hilbert space of histories, on which the Poisson lifting is an isometry. This closes a circle. A lifted history is precisely an initial datum for the singular Schrödinger equation with nonlinear Neumann interaction of our companion paper, whose well-posedness theory therefore transfers to $\mathcal{L}^su=μ|u|^{p-1}u$ with prescribed past. With our earlier work on the Bessel operator on a half-line, the three papers form a single programme, of which the present one is the closing step.
