## 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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