Explicit Spectral formulae for scaling quantum graphs

Abstract

We present an exact analytical solution of the spectral problem of quasi one-dimensional scaling quantum graphs. Strongly stochastic in the classical limit, these systems are frequently employed as models of quantum chaos. We show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form $E_n=f(n)$, where $n$ is the sequence number of the energy level of the quantum graph and $f$ is a known function, which depends only on the physical and geometrical properties of the quantum graph. Our method of solution motivates a new classification scheme for quantum graphs: we show that each quantum graph can be uniquely assigned an integer $m$ reflecting its level of complexity. We show that a taut string with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs.