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 Pnn>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 ⊂C 1(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 L loc∞(R;L2,2(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 H 1(R3). © 1993 American Institute of Physics.

Convergence of Nelson diffusions with time-dependent electromagnetic potentials / A. Posilicano, S. Ugolini. - In: JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 0022-2488. - 34:11(1993), pp. 5028-5036.

Convergence of Nelson diffusions with time-dependent electromagnetic potentials

S. Ugolini
Ultimo
1993

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 Pnn>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 ⊂C 1(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 L loc∞(R;L2,2(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 H 1(R3). © 1993 American Institute of Physics.
stochastic variational-principles; mechanics
Settore MAT/07 - Fisica Matematica
Settore MAT/06 - Probabilita' e Statistica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/380745
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