Ergodic Schrödinger Operators
Ergodic Schrödinger Operators
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DOI：
10.1007/978-1-4684-9172-2_8
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
P. Bougerol;J. Lacroix
中科院分区：
文献类型：
--
作者：
P. Bougerol;J. Lacroix
我们现在假设(An，Bn)，nεZz是平稳随机过程。这意味着实随机变量(An(ω)，Bn(ω))定义在某个完备的概率空间(Ω，a，ℙ)上，并且存在Ω的可逆可测变换ℙ，使得An+1=An：θ，Bn+1，：θ。通常，我们不编写变量ω，并且当一个属性只依赖于序列(an，bn)的常见规律时，我们省略索引n，而谈论变量(A)和(B)。我们称H上的伴随算子族H(ω)是遍历的，如果θ不变的Ω可测子集为零或一个ℙ测度。很容易看出，H°θ=U-1HU，其中U是移位度量。很容易看到Hoθ=U-1HU，其中U是ZZ上的移位算子，(Uψ)n=ψn-1。
We now suppose that (an,bn), n ε ZZ, is a stationary random process This means that the real random variables (an(ω), bn(ω)) are defined on some complete probability space (Ω,a,ℙ) and that there exists an invertible measurable transformation θ of Ω, leaving ℙ invariant and such that an+1=anºθ, bn+1, ºθ. In general we don’t write the variable ω and when a property depends only upon the common law of the sequence (an, bn) we omit the index n and speak of the variables (a) and (b). We say that the family H(ω) of associated operators on H is ergodic if a θ invariant measurable subset of Ω is of zero or one ℙ measure. It’s easily seen that H° θ = U-1H U where U is the shift measure. It’s easily seen that H o θ = U-1H U where U is the shift operator on ZZ, (Uψ)n= ψn-1.