An extension theory for fully fractional Schr\"odinger equations with memory

Abstract

We develop a Caffarelli-Silvestre extension theory for the fully fractional Schr\"odinger operator $\mathcal{L}^s=(\partial_t-i\Delta_x)^s$, $0<s<1$, 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 $e^{i\frac{\pi s}{2}}$; 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 Schr\"odinger equation with nonlinear Neumann interaction of our companion paper, whose well-posedness theory therefore transfers to $\mathcal{L}^su=\mu|u|^{p-1}u$ 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.