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Probability conserving elliptic operators. (Russian) Zbl 0661.35022
The main result of this work is as follows: Let \(a_{kj}(x)\in C^{\infty}({\mathbb{R}}^ e)\) \((k,j=1,...,l)\) be such real functions on \({\mathbb{R}}^ l\), that \(a_{kj}(x)=a_{jk}(x)\) \(\forall k,j=1,...,l\), \(\forall x\in {\mathbb{R}}^ l\), \(0<\sum_{k,j}a_{kj}(x)\xi_ k\xi_ j\) \((\forall x\in {\mathbb{R}}^ l)\) \((\forall \xi \in {\mathbb{R}}^ l\), \(\xi\neq 0)\) \((\exists R>0)\) \(\sum_{k,j}a_{kj}(x)x_ kx_ j/| x|^ 2\leq const | x|^ 2\ln | x|\) \((\forall x:| x| >R)\). Then for all \(\lambda >0\) the operator (\(\lambda\) I- \(\sum_{j,k}(\partial /\partial x_ k)a_{kj}(\partial /\partial x_ j))\upharpoonright C_ 0^{\infty}({\mathbb{R}}^ l)\) has a dense range in the space \(L^ 1({\mathbb{R}}^ l)\) and (1) \(e^{-tH_ 1}=1\) where \(H_ 1\) is the closure of \(-\sum^{l}_{k,j=1}(\partial /\partial x_ k)a_{kj}\partial /\partial x_ j\) in \(L^ 1({\mathbb{R}}^ l).\)
The property (1) means that the operator \(H_ 1\) conserves the probability.
Reviewer: V.S.Rabinovich

35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
60J60 Diffusion processes