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This work is devoted to non-linear stochastic Schr\"odinger equations withmultiplicative fractional noise, where the stochastic integral is definedfollowing the Riemann-Stieljes approach of Z\"ahle. Under the assumptions thatthe initial condition is in the Sobolev space $H^q(\Rm^n)$ for a dimension $n$less than three and $q$ an integer greater or equal to zero, that the noise isa $Q-$fractional Brownian motion with Hurst index $H\in(1/2,1)$ and spatialregularity $H^{q+4}(\Rm^n)$, as well as appropriate hypotheses on thenon-linearity, we obtain the local existence of a unique pathwise solution in$\calC^0(0,T,H^q(\Rm^n)) \cap \calC^{0,\gamma}(0,T,H^{q-2}(\Rm^n))$, for any$\gamma \in [0,H)$. Contrary to the parabolic case, standard fixed pointtechniques based on the mild formulation of the SPDE cannot be directly usedbecause of the weak smoothing in time properties of the Schr\"odingersemigroup. We follow here a different route and our proof relies on a change ofphase that removes the noise and leads to a Schr\"odinger equation with amagnetic potential that is not differentiable in time.

A note on stochastic Schr\"odinger equations with fractional multiplicative noise