Abstract
Some recent results on the convergence of Nelson diffusions are extended to the case of Schrödinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrödinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;ℛ×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in ℛ, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrödinger equations are continuous in the (time-dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).
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