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Zbl 0755.47050
Cheremshantsev, S.E.
Hamiltonians with zero-range interactions supported by a Brownian path. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 56, No.1, 1-25 (1992). ISSN 0246-0211

Let $\Omega$ be a closed subset of Lebesgue measure 0 in $R\sp m$, $X$ a space with finite measure $\mu$, and $\omega$ a measurable map of $X$ with range $\Omega$. Let $F$ be a real bounded measurable function on $X$. The author describes a general scheme to construct selfadjoint Hamiltonians $$H=-\Delta\sb x+\int\sb X d\mu(t)\delta(x- \omega(t))F(t).\tag1$$ He uses a limiting procedure with ultraviolet cut- off in Fourier-representation, and the Trotter-Kato theorem on the convergence of the sequence of selfadjoint operators in the strong resolvent (S.R.) sense. First of all he gives the following two conditions of $\omega(t)$;\par A1. There exists $\delta\in(0,{1\over2})$ such that $$\int\sb{X\sp 2}d\mu(t)d\mu(s)\vert\omega(s)-\omega(t)\vert\sp{-2\delta}<+\infty$$ holds.\par A2. $N(t,u)=\int\sb X d\mu(s)(1+\vert\omega(s)-\omega(u)\vert\sp{- 1})(1+\vert \omega(s)-\omega(t)\vert\sp{-1})\in L\sb 2(X\sp 2)$.\par In Theorems 1 and 2 he treats the cases without renormalization and with renormalization, respectively.\par Theorem 1. $\omega$ satisfies A1 for $m=2$ and A2 for $m=3$. Then a sequence of selfadjoint operators $\{H\sb n\}$ given by cut-off converge to $H$ in the S.R. sense.\par Theorem 2 treats $m=4$ (A1) or $=5$ (A2) and gives similar results. His results can be applied to the case where $\Omega$ is a Brownian path in $R\sp m$, $m=3,4$ and 5.
[H.Yamagata (Osaka)]

MSC 2000:: *47N50 Appl. of operator theory in quantum physics
81Q10 Selfadjoint operator theory in quantum theory

Keywords: measurable map; selfadjoint Hamiltonians; ultraviolet cut-off in Fourier- representation; Trotter-Kato theorem

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