LETTER TO THE EDITOR Green function approach for general quantum graphs

We present a recursive prescription to calculate the exact Green function for general quantum graphs. For the closed case, the expression for the poles of G—which gives the individual eigenstates—has the structure of a semiclassical formula, where the sum over the periodic orbit is already performed. As applications we discuss eigenstate localization for a three-arm closed star and filter-like mechanisms for transmission throughout an open trident graph. PACS number: 03.65.Nk A graph is a network of M connected vertices V, figure 1. Each Vm is attached to Nm arms of length mn (n = 1,..., Nm). If both ends (just one end) of an arm are fixed to vertices it is called a bond (semi-infinite lead). There are no leads in closed graphs. Along any arm, ψ(x) is defined uniquely from a 1D Schrödinger equation. The total wavefunction is given by such piecewise solutions properly matched at the vertices [1]. For a long time quantum graphs have been used in the description of real system