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Existence of the exponentially localized Wannier functions. (English) Zbl 0545.47012
Let $$\Omega =\{z=(z_ i)^ N_{i=1}\in {\mathbb{C}}^ N| | Im z_ i|<a\}$$, H a Hilbert space and let $$P:\Omega \to {\mathcal L}(H)$$ be an analytic projection-valued function such that $$P(z)^*=P(z), z\in {\mathbb{R}}^ N,$$ $$P(z)=P(z+2\pi p),$$ $$p\in {\mathbb{Z}}^ N$$, $$z\in \Omega$$. Does there exist an operator-valued analytic function $$A:\Omega \to G{\mathcal L}(H)$$ such that $$P(z)=A(u)P(0)A(z)^{-1}, A(0)=I, A(z)^{- 1}=A(z)^*$$, $$z\in {\mathbb{R}}^ N$$ and $$A(z)P(0)=A(z+2\pi p)P(0), z\in \Omega$$, $$z\in {\mathbb{Z}}^ N?$$ Without the periodicity conditions on P and A this problem is fully solved [see e.g., T. Kato, Perturbation theory of linear operators (1976; Zbl 0342.47009) II-§ 4.2,6.2] and for $$N=1$$ it is also solved. In the present paper the existence of A is proved for $$N>1$$ under the following conditions on P:
(i) dim R(P(z))$$=1,$$
(ii) there exists an antilinear involution $$\theta: H\to H$$ such that $$\theta P(z)\theta =P(-z)$$, $$z\in {\mathbb{R}}^ N.$$
The result is used to show the existence of exponentially localized Wannier functions.
Reviewer: G.P.A.Thijsse

##### MSC:
 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) 47A55 Perturbation theory of linear operators
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