Periodic orbit theory and the statistical analysis of scaling quantum graph spectra.

The explicit solution to the spectral problem of quantum graphs found recently by Dabaghian and Blümel [Phys. Rev. E 68, 055201(R) (2003); 70, 046206 (2004); JETP Lett. 77, 530 (2003)] is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations sn = kn - kn-1, and the distribution of the spectral oscillations around the average, deltakn=kn - kn.

Explicit spectral formulas for scaling quantum graphs.

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 network of taut strings with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs.

Explicit analytical solution for scaling quantum graphs.

We show that scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos and quantum chaotic systems are not usually explicitly analytically solvable.

Explicitly solvable cases of one-dimensional quantum chaos.

We identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact, convergent periodic orbit expansions for individual energy levels, thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete, explicit, and exact.