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On the coincidence of the spectra of random elliptic operators. (English. Russian original) Zbl 0568.35091

Math. USSR, Sb. 51, 455-471 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 4, 460-476 (1984).
Let \(\Omega\) be a probability space on which acts an n-dimensional measure preserving ergodic dynamical system T(x), \(x\in {\mathbb{R}}^ n\), with an associated n-parameter group of unitary operators \(U_ x\) in \(L^ 2(\Omega)\) defined by \((U_ xt)(\omega)=f(T(x)\omega),\) \(f\in L^ 2(\Omega)\). Define \(C_ b^{\infty}(\Omega)\) as the set of functions \(f\in L^{\infty}(\Omega)\) for which the function f(T(x)\(\omega)\) and all of its derivatives in x are bounded, uniformly in \(\omega\). Let \(C_ b^{\infty}(R^ n)\) consist of all functions g in \(C^{\infty}({\mathbb{R}}^ n)\) with all derivatives bounded. The expression \[ A_{\omega} u(x)=\sum_{| \alpha | \leq m}a_{\alpha}(T(x)\omega) D_ x^{\alpha}u(x), \] where \(u\in C_ b^{\infty}({\mathbb{R}}^ n)\), defines a random differential operator in \(L^ 2({\mathbb{R}}^ n)\); here \(\alpha =(\alpha_ 1,...,\alpha_ n)\) is a multi-index, \(D_ x^{\alpha}=D_ 1^{\alpha_ 1}...D_ n^{\alpha_ n},\quad D_ j=i^{-1}\partial /\partial x_ i,\quad | \alpha | =\alpha_ 1+\alpha_ 2+...+\alpha_ n,\) \(\omega\in \Omega\), and \(a_{\alpha}\in C_ b^{\infty}(\Omega)\) for each multi- index \(\alpha\). Also, the expression \(Af=\sum_{| \alpha | \leq m}a_{\alpha}(\omega) D^{\alpha}f,\) \(f\in C_ b^{\infty}(\Omega)\) defines an operator in \(L^ 2(\Omega)\), where \(D^{\alpha}=i^{-| \alpha |}\partial_ 1^{\alpha_ 1}...\partial_ n^{\alpha_ n},\) and the derivatives \(\partial_ i\) are defined by \((\partial_ if)(\omega)=(\partial /\partial x_ i) f (T(x)\omega)|_{x=0}.\) The authors consider several results relating the spectral sets \(\sigma\) (A) and \(\sigma (A_{\omega})\), \(\omega\in \Omega\). Let \(\Gamma\) denote the group of periods of the unitary group \(\{U_ x: x\in {\mathbb{R}}^ n\}\) (i.e. \(\Gamma =\{y\in {\mathbb{R}}^ n: U_ yg=g\), all \(g\in L^ 2(\Omega)\}).\)
It is shown that if A, \(A_{\omega}\), \(\omega\in \Omega\), are elliptic and self-adjoint, \(\sigma (A)\subset \sigma (A_{\omega})\) and that if \(\Gamma =\{0\}\), then \(\sigma (A)=\sigma (A_{\omega})\) for almost all \(\omega\in \Omega\). If T(x) is strictly ergodic (i.e. the Borel normalized measure invariant relative to T(x) is unique) the following result is also given. Theorem: Let \(\Gamma =\{0\}\) and let \(A_{\omega}\) be random elliptic selfadjoint operators of order \(m>0\) with symbols \(a_{\omega}(x,\xi)\) such that the function \(\omega \to \partial^{\alpha}_{\xi} \partial_ x^{\beta} a_{\omega}(x,\xi)\) belongs to C(\(\Omega)\) for all fixed \(\alpha\), \(\beta\), x, \(\xi\). Then \(\sigma (A)=\cap_{\omega \in \Omega}\sigma (A_{\omega})\).
Reviewer: I.Knowles

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35J25 Boundary value problems for second-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
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