Analytic structure of Bloch functions for linear molecular chains

In this talk I will discuss Hamiltonians of the form $H=-{\bf \nabla}^2+v(x,y,z)$, with $v (x,y,z)$ periodic along the $z$ direction, $v(x,y,z+b)=v(x,y,z)$. The wavefunctions of $H$ are the well known Bloch functions $\psi_{n,\lambda}(x,y,z)$, with the fundamental property $\psi_{n,\lambda}(x,y,z+b)=\lambda \psi_{n,\lambda}(x,y,z)$ and $\partial_z\psi_{n, \lambda}(x,y,z+b)=\lambda \partial_z\psi_{n,\lambda}(x,y,z)$. I will give the generic analytic structure (i.e. the Riemann surface) of $\psi_{n,\lambda}(x,y,z)$ and their corresponding energy, $E_n (\lambda)$, as functions of $\lambda$. I will also discuss several applications, like a compact expression of the Green's function or the asymptotic behavior of the density matrix and other correlation functions for insulating molecular chains.