An extension theory for fully fractional Schrödinger equations with memory

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 positivi