Abstract
Under mild condition on the modulus φ = ∣ ψ∣ of the time independent wave function ψ, we prove that the generalized Schrödinger operator ℒφ= ℒ + 2 Γ (φ, ·)/φ (or the generator of Nelson's diffusion) defined on a good space of test-functions \(D\)on a general Polish space, generates a unique semigroup of class (C o) in L 1. This result reinforces the known results on the essential Markovian self-adjointness in different contexts and extends our previous works in the finite dimensional Euclidean space setting. In particular it can be applied to the ground or excited state diffusion associated with an usual Schr\"odinger operator \( - L + V\), and to stochastic quantization of several Euclidean quantum fields.
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Wu, L. Uniqueness of Nelson's Diffusions II: Infinite Dimensional Setting and Applications. Potential Analysis 13, 269–301 (2000). https://doi.org/10.1023/A:1008707703191
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DOI: https://doi.org/10.1023/A:1008707703191