Schroedinger Equations with Fractional Laplacians

It is shown that the unique solution of ∂/∂t Ψ(t,x) = -(z2)α/2(-Δ)α/2 Ψ(t,x)+V(z,x)Ψ(t,x), Ψ(0,x)=f(x), } can be represented as { Ψ(t,x) = Ef(x+(z)1/αXs) exp{∫0tV(z,x+(z)1αXu)du}, } where X=(Xt , t≥ 0) is a stable process whose generator is (-Δ)α/2 with X0=0