# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Wannier, G.H.: The structure of electronic excitation levels in insulating crystals. Phys. Rev.52, 191-197 (1937) · Zbl 0017.23601 [2] Callaway, J.: Quantum theory of the solid state. New York: Academic Press 1974 [3] Des Cloizeaux, J.: Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys. Rev.135A, 685-697 (1964) [4] Des Cloizeaux, J.: Analytical properties ofn-dimensional energy bands and Wannier functions. Phys. Rev.135A, 698-707 (1964) [5] Nenciu, A., Nenciu, G.: Dynamics of Bloch electrons in external electric fields: II. The existence of Stark-Wannier ladder resonances. J. Phys. A15, 3313-3328 (1982) [6] Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev.115, 809-821 (1958) · Zbl 0086.45101 [7] Blount, I.E.: Solid State Phys.13, 305-373 (1962) [8] Kato, T.: Perturbation theory of linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601 [9] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. IV. New York, San Francisco, London: Academic Press 1978 · Zbl 0401.47001 [10] Krein, S.G.: Linear differential equations in Banach spaces. Moskva: Nauka 1967 · Zbl 0193.09302 [11] Wasow, W.: Topics in the theory of linear ordinary differential equations having signularities with respect to a parameter. Lecture Notes, IRMA Univ. Louis Pasteur, Strasbourg 1977 [12] Sibuya, Y.: Math. Ann.161, 67-77 (1965) · Zbl 0229.15011 [13] Hsieh, P-F., Sibuya, Y.: J. Math. Anal. Appl.14, 332-340 (1966) · Zbl 0143.05105 [14] Kittel, C.: Quantum theory of solids. New York: Wiley 1963 · Zbl 0121.44701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.