Existence of the exponentially localized Wannier functions
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., \textit{T. Kato}, Perturbation theory of linear operators (1976; Zbl 0342.47009) II-{\S} 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.
existence of exponentially localized Wannier functions
scientific article; zbMATH DE number 3868052
scientific article