An extension theory for fully fractional Schrödinger equations with memory

We study nonlinear Schrödinger equations with memory through an extension theory for the fully fractional Schrödinger operator $$({\mathrm{\partial }}_t-i{\mathrm{\Delta }}_x{)}^s,0<s<1.$$ Because the operator is nonlocal in both space and time, its natural Cauchy problem is posed with a prescribed past history rather than a classical initial condition. We construct a semigroup definition of $({\mathrm{\partial }}_t-i{\mathrm{\Delta }}_x{)}^s$, derive an equivalent local extension problem in one additional spatial variable, and compute the corresponding oscillatory Poisson kernel explicitly. The extension yields a Poisson lifting of history data and an intrinsic quadratic form on histories, establishing an explicit identity between the weighted bulk energy of the lifting and a nonlocal energy determined entirely by the prescribed past evolution. As an application, and in combination with the nonlinear boundary-interaction theory developed in our companion paper, we obtain a well-posedness theory for the associated nonlinear memory problem. The well-posedness is established in an extension-induced mild sense: for a general history in the natural energy space the future evolution is defined as the boundary trace of the corresponding solution in the half-space, and the fractional equation itself is recovered, in $L^2$ and for positive times, under an additional graph-domain hypothesis.