No Quantum Ergodicity for Star Graphs

We investigate statistical properties of the eigenfunctions of the Schrödinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These structures are analogous to scars on short unstable periodic orbits.