Explicit spectral formulas 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 = f͑n͒, 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.

Analytic description of statistics of spectra of quantum graphs

“…where the maximum frequency L 0 is given by the sum of the optical lengths of the graph edges [9]- [15]. This representation allows classifying spectral determinants [13]- [15] in accordance with the quantity in the right-hand side of Eq.…”

“…The size 2N P × 2N P of the matrix T (E) is therefore determined by the number of graph edges. For example, for the two-edge hydra with the edge lengths l 1 and l 2 [8]- [15], the PSS consists of the three vertices of the graph, V 1 , V 2 , and V 3 . Taking the directions of the orbit intersections with the PSS and the signs of the scattering coefficients [8] into account and following the rules for constructing the operator T (k), we obtain …”

“…Nevertheless, as shown in [13]- [15], quantum graphs provide an example of a system in which such a set of separating points k n , n = 1, 2, . .…”

“…In what follows, we discuss the exact solution of the spectral problem proposed in [13]- [15] and derive a statistical description of spectral characteristics using this solution.…”

Dressed Quantum Graphs With Optical Nonlinearities Approaching the Fundamental Limit

“…At the same time, multi-electron systems with arbitrary interactions yield the same universal results as one-electron systems 37 , suggesting that the single electron models are suitable vehicles for studying the relationships of graph geometry and topology to the scaling properties of the nonlinear optical tensors. The single electron quantum graph is a well-studied, exactly solvable model of quantum chaos 38,39,40,41,42,43,44,45 . With this in mind, we initiated our studies of the elementary QG model for nonlinear optics by focusing first on undressed edges and calculated the off-resonance first (β ijk ) and second (γ ijkl ) hyperpolarizability tensors (normalized to their maximum values) of elementary graphical structures, such as wires, closed loops, and star vertices 34,46 and to investigate the relationship between the topology and geometry of a graph and its nonlinear optical response through its hyperpolarizability tensors 35 .…”