## 3 Real Wannier Functions and Dispersive Properties
Also, for every \(n\in\mathbb{N}\), and \(k\in[0,\pi/b]\), there exists a unique \(E\in B_{n}\) satisfying (60). We denote such \(E\in B_{n}\) by \(E_{k,n}\). The large \(E\) behavior of \(\mu(E)\) also implies that there exist \(c\), \(C>0\) such that \(\varepsilon_{n}=\min B_{n}\) satisfies \(cn^{2}\leq\varepsilon_{n}\leq Cn^{2}\), \(\forall n\in\mathbb{N}\). The solutions of (60) can be then parametrized as \(E_{n,k}\), \(k\in[0,\pi/b]\), \(n\in\mathbb{N}\), and \(B_{n}=[E_{n,\pi/b},E_{n,0}]\). Also, we let \(E(-k,n)=E_{k,n}\) and extend \(E_{n,k}\) to \(k\in\mathbb{R}\) by \(2\pi/b-\)periodicity. This notation is consistent with (60). For \(V\) non-negative all bands belong to \(\overline{\mathbb{R}^{+}}\). By the implicit function theorem, given \(n\in\mathbb{N}\), \(E_{n,k}\) is real analytic for \(k\in(0,\pi/b)\), and is continuous in \([0,\pi/b]\). The even and \(2\pi/b-\)periodic extension of \(E_{n,k}\) to real \(k\) is continuous in \(\mathbb{R}\), and real analytic at all points outside the lattice \(\mathbb{Z}\frac{\pi}{b}\), for all \(n\in\mathbb{N}\). Regularity of \(E_{n,k}\) at points \(\mathbb{Z}\frac{\pi}{b}\) for given \(n\) follows under gap conditions for the edges of the band \(B_{n}\). Consider now \(E=E_{n,k}\) as above a solution of (60) for some \(n\in\mathbb{N}\), \(k\in[0,\pi/b]\), and the corresponding real solutions \(\Psi_{1}(\cdot;E)\), \(\Psi_{2}(\cdot;E)\). Solving (56), (57) we have  
\[\frac{A}{B}=\frac{\Psi_{2}(b;E)}{\lambda-\Psi_{1}(b;E)}, \tag{61}\]  
and we obtain  
\[\Psi(y;E)=A\bigg{(}\Psi_{1}(y;E)+\frac{\Psi_{2}(b;E)}{\lambda-\Psi_{1}(b;E)} \Psi_{2}(y;E)\bigg{)}, \tag{62}\]  
with \(\lambda=e^{ikb}\), \(E=E_{n,k}\), \(n\in\mathbb{N}\), \(k\in(0,\pi/b)\). The expression can be extended to the endpoints \(k=0\), \(\pi/b\), under conditions we discuss below. Also, the complex coefficient \(A\) is free, e.g., it can be also chosen to normalize \(\Psi(y;E)\). In general it may depend on \(n\), \(k\), and we write \(A=A_{n,k}\). Denote \(\Psi(\cdot;E)=\Psi(\cdot;E_{n,k})\) by \(\Psi_{n,k}\), \(n\in\mathbb{N}\), \(k\in[0,\pi/b]\). Clearly \(\Psi_{n,k}^{*}\) is also a solution of (51). We then let  
\[\phi_{n,k}=\left\{ \begin{array}{ll}\Psi_{n,k}&\text{ if }\quad 0<k<\frac{\pi}{b}, \\ \Psi_{n,-k}^{*}&\text{ if }\quad-\frac{\pi}{b}<k<0\end{array} \right.. \tag{63}\]  
The functions \(\phi_{n,k}\) are extended by \(2\pi/b-\)periodicity for \(k\in\mathbb{R}\setminus\mathbb{Z}\frac{\pi}{b}\) and are Bloch functions, see (12), (13), (14). We check that the corresponding Wannier functions are real. By (18), it suffices to show that the \(w_{n}^{0}\) are real. By (16)  
\[w_{n}^{0}(y)=\sqrt{\frac{b}{2\pi}}\int_{-\frac{\pi}{b}}^{\frac{\pi}{b}}\phi_{n,k}(y)\,dk, \tag{64}\]  
and by (63)  
\[\int_{-\frac{\pi}{b}}^{\frac{\pi}{b}}\phi_{n,k}(y)dk=\int_{-\frac{\pi}{b}}^{0} \Psi_{n,-k}^{*}(y)\,dk+\int_{0}^{\frac{\pi}{b}}\Psi_{n,k}(y)dk=2\text{Re}\int _{0}^{\frac{\pi}{b}}\Psi_{n,k}(y)\,dk, \tag{65}\]  
\(\forall y\in\mathbb{R}\). It is assumed that the last integral is well defined. We now give a condition that makes the above construction well defined, leading to \(w_{n}^{0}\in L^{2}(\mathbb{R},\mathbb{R})\), for all \(n\in\mathbb{N}\). In particular, assume that the limits  
\[\lim_{k\to 0^{+}}\frac{dE_{n,k}}{dk},\quad\lim_{k\to\pi/b^{-}}\frac{dE_{n,k}}{dk}\]exist (and are finite). Then the corresponding limits of the fraction \(\Psi_{2}(b;E)/(\lambda-\Psi_{1}(b;E))\) also exist, and \(\Psi_{n,k}\) given by the right hand side of (62), with \(A=1\), is well defined for all \(n\in\mathbb{N}\), \(k\in[0,\pi/b]\). We further see that the \(\Psi_{n,\cdot}(\cdot)\) are continuous in \([0,\pi/b]\times\mathbb{R}\), \(\forall n\in\mathbb{N}\). Defining \(\phi_{n,k}\) by (63), the integrals in (64), (65) are finite for all \(y\in\mathbb{R}\). It follows that \(\phi_{n,\cdot}(y)\in L^{2}([-\pi/b,\pi/b];\mathbb{C})\), for all \(n\in\mathbb{N}\), \(y\in\mathbb{R}\).