We show that all scaling quantum graphs are explicitly integrable, i.e., that any one of their spectral eigenvalues E_n is computable analytically, explicitly, and individually for any given n. This is surprising, since quantum graphs are excellent models of quantum chaos (see, e.g., T. Kottos and H. Schanz, Physica E 9, 523 (2001)).

This Letter discusses the question of how the white noise of radiative corrections affects the chaotic properties of an original quantum system. It is shown, by an explicit mathematical analysis, that the radiative corrections, in effect, remove the original chaos in the system.

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.

Using quantum graph theory we establish that the ray-splitting trace formula proposed by Couchman et al. (Phys. Rev. A 46, 6193, 1992) is exact for a class of one-dimensional ray-splitting systems. Important applications in combinatorics are suggested.

We present quantum graphs with remarkably regular spectral characteristics. We call them regular quantum graphs. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions for the spectrum of regular quantum graphs in the form of explicit and exact periodic orbit expansions for each individual energy level

It is shown that a recently discovered representation of the Green's function is equivalent to a certain "dynamical ansatz" for the corresponding path integral, which brings about a convenient method of nonperturbative approximations. Based on this observation, a set of nonperturbative approximations to the trace of the Green's function is established.

Explicit, exact periodic orbit expansions for individual eigenvalues exist for a subclass of quantum networks called regular quantum graphs. We prove that all linear chain graphs have a regular regime.

We consider a class of simple quasi-one-dimensional classically nonintegrable systems that capture the essence of the periodic orbit structure of general hyperbolic nonintegrable dynamical systems. Their behavior is sufficiently simple to allow a detailed investigation of both classical and quantum regimes. Despite their classical chaoticity, these systems exhibit a “nonintegrable analogue” of the Einstein–Brillouin–Keller quantization formula that provides their spectra explicitly, state by state, by means of convergent periodic orbit expansions

A general analytical approach to the statistical description of quantum graph spectra based on the exact periodic orbit expansions of quantum levels is discussed. The exact and approximate expressions obtained in [5] for the probability distribution functions using the spectral hierarchy method are analyzed. In addition, the mechanism of appearance of the universal statistical properties of spectral fluctuations of quantum-chaotic systems is considered in terms of the semiclassical theory of periodic orbits.

A cycle expansion technique for discrete sums of several PF operators, similar to the one used in the standard classical dynamical z -function formalism is constructed. It is shown that the corresponding expansion coefficients show an interesting universal behavior, which illustrates the details of the interference between the particular mappings entering the sum.