Abstract
Let ψt, ψ nt ,n≧1, be solutions of Schrödinger equations with potentials form-bounded by −1/2δ and initial data inH 1(ℝd). LetP, P n,n≧1, be the probability measures on the path space Ω=C(ℝ+,ℝd) given by the corresponding Nelson diffusions. We show that if {ψ nt } n≧1 converges to ψt inH 1(ℝd), uniformly int over compact intervals, then
converges to
in total variation ∀t≧0. Moreover, if the potentials are in the Kato classK d , we show that the above result follows fromH 1-convergence of initial data, andK d -convergence of potentials.
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Dell'Antonio, G., Posilicano, A. Convergence of Nelson fiffusions. Commun.Math. Phys. 141, 559–576 (1991). https://doi.org/10.1007/BF02102816
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DOI: https://doi.org/10.1007/BF02102816