Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields |
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| Authors: | Sergio Albeverio Jens-Erik Fenstad Raphael Høegh-Krohn Witold Karwowski Tom Lindstrøm |
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| Affiliation: | 1. Mathematisches Institut, Ruhr-Universität, 4630 Bochum 1, W. Germany;2. Matematisk Institutt, Universitetet i Oslo, Oslo, Norway;3. Department of Mathematics, Stanford University, Stanford, CA, USA;4. Dep. Physique Théorique, Université de Provence and Centre de Physique Théorique, CNRS, Marseille, France;5. Inst. Theoretical Physics, University of Wroc?aw, Wroc?aw, Poland;6. Matematisk Institutt, Norges Tekniske Högskole, Trondheim, Norway |
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| Abstract: | We discuss hamiltonians in L2(Rd, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫01λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?d4-type is also discussed. The proofs use methods of nonstandard analysis. |
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