## 3 Real Wannier Functions and Dispersive Properties  
The Hamiltonion structure of the infinite system for the Wannier coefficients (30) assumed real functions. In this section we describe the construction of real Wannier functions using explicit constructions of the Bloch functions. We also observe that the linear part of (30) will in general couple modes from different bands. This is a main difference between our system and the equations considered in [11; 12; 26; 27]. We show that we can still however diagonalize the system using trigonometric functions. In that sense, the Wannier-Bloch analysis leads to a homogenized, i.e., effectively translation invariant, linear part, see [29; 30]. To construct the Wannier functions we examine the Schrodinger equation  
\[-\partial_{y}^{2}\Psi+V(y)\Psi=E\Psi,\quad\Psi:\mathbb{R}\to\mathbb{C} \tag{51}\]  
with \(V\) nonegative and \(b\)-periodic as a second order ODE with a real parameter \(E\in\mathbb{R}\). We assume that \(V\) is also piecewise Lipschitz. The spectrum of \(-\partial_{y}^{2}+V\) is given by the set of real \(E\) for which all solutions of (51) are bounded, see e.g., [24; 35]. The equations for the real and imaginary parts of \(\Psi\) decouple, and all complex valued solutions are of the form \(A\Psi_{1}(;E)+B\Psi_{2}(;E)\), \(A\), \(B\in\mathbb{C}\), where \(\Psi_{1}(;E)\), \(\Psi_{2}(;E)\) are any two linearly independent real solutions. We consider solutions \(\Psi_{1}(;E)\), \(\Psi_{2}(;E)\) with initial conditions  
\[\Psi_{1}(0;E) =1, \Psi_{1}^{\prime}(0;E) =0, \tag{52}\] \[\Psi_{2}(0;E) =0, \Psi_{2}^{\prime}(0;E) =1. \tag{53}\]  
By the Hamiltonian structure of (51), seen a non-autonomous ODE on the plane, the corresponding solutions are linearly independent, i.e., \([\Psi_{1}(y;E),\Psi_{1}^{\prime}(y;E)],[\Psi_{2}(y;E),\Psi_{2}^{\prime}(y;E)]\) are linearly independent since the (linear) solution map is symplectic, \(\forall y\in\mathbb{R}\). Given \(E\in\mathbb{R}\), a solution \(\Psi(;E)\) is bounded if and only if there exists \(\lambda\in\mathbb{C}\), \(|\lambda|=1\), for which  
\[\Psi(y+b;E)=\lambda\Psi(y;E),\quad\forall y\in\mathbb{R}, \tag{54}\]  
see [21]. Then we must also have  
\[\Psi^{\prime}(y+b;E)=\lambda\Psi^{\prime}(y;E),\quad\forall y\in\mathbb{R}. \tag{55}\]  
Let \(\Psi(y;E)=A\Psi_{1}(y;E)+B\Psi_{2}(y;E)\) for \(A\), \(B\) complex, then (54), (55) at \(y=0\) and (52), (53) imply the system  
\[A\Psi_{1}(b;E)+B\Psi_{2}(b;E) = \lambda A \tag{56}\] \[A\Psi_{1}^{\prime}(b;E)+B\Psi_{2}^{\prime}(b;E) = \lambda B. \tag{57}\]  
Then \(\lambda\) must be an eigenvalue of the matrix \(M\) defined by  
\[M=\bigg{(} \begin{array}{cc}\Psi_{1}(b;E)&\Psi_{2}(b;E)\\ \Psi_{1}^{\prime}(b;E)&\Psi_{2}^{\prime}(b;E)\end{array} \bigg{)}. \tag{58}\]  
If \(\lambda\) is an eigenvalue of \(M\), and \(\lambda=e^{ikb}\), then \(k\) and \(E\) are related to  
\[\mu(E)=\cos kb, \tag{59}\]  
where  
\[\mu(E)=\frac{\Psi_{1}(b;E)+\Psi_{2}^{\prime}(b;E)}{2}, \tag{60}\]  
see [21]. The dependence of \(\mu\) on the choice of \(\Psi_{1}\), \(\Psi_{2}\), i.e., the initial conditions (52), (53) is supressed from the notation. We recall some properties of \(\mu(E)\), and the solutions of (59), see [21; 22; 23; 24]. The function \(\mu(E)\), \(\mu\) is entire and \(\mu^{\prime}(E)=0\) implies \(|\mu(E)|\geq 1\). Also, \(\mu(E)\to\infty\) as \(E\to-\infty\), and \(\mu(E)\to\cos(b\sqrt{E})\) as \(E\to\infty\). Also \(\mu(E)\to\infty\) as \(E\to-\infty\). For \(V\) non-negative \(\mu\) has no negative critical points and an infinite set of positive critical points that become equidistant. A band is a maximal connected interval of \(\mathbb{R}\) where \(\mu(E)\) is monotone and satisfies \(|\mu(E)|\leq 1\). By the above, there is an infinite number of bands \(B_{n}\), \(n\in\mathbb{N}\), and a natural way to enumerate them so that points \(n<n^{\prime}\), \(E\in B_{n}\), \(E^{\prime}\) implies \(E_{n}\leq E_{n^{\prime}}\), with equality holding for \(n=n+1\) and \(E=\max B_{n}\), \(E^{\prime}=\min B_{n^{\prime}}\). Every \(E\) satisfying (60) must belong to exactly one band.