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

Abstract

The explicit solution to the spectral problem of quantum graphs found recently in [20], is used to produce the exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations, s n = k n − k n−1 , and the distribution of the spectral oscillations around the average, δk n = k n −k n .

Аналитическое Описание Статистики Спектров Квантовых Графов

“…рассматривавшиеся в работе [35]. В [13]- [15] было также показано, что в силу определенных аналитических свойств экспоненциальной суммы вида (11) [15], [36], [37] в качестве k…”

“…в котором x -это набор из N P − 1 независимых равномерно распределенных случайных величин, а коэффициенты C Распределение значений δf (0) = δf (0) x в таком случае получается из выражения [35], [40]…”

“…Как показывают численные расчеты, распределение флуктуаций δ n , описываемых суммой (26), для достаточно сложных графов (например, для шестивершинной Ю. А. ДАБАГЯН гидры или полностью связного тетраэдра) хорошо приближается распределением Гаусса со среднеквадратичным отклонением [35], [41], [49], [50]…”

Complexity of spectral sequences: semiclassical approach

“…where η (j+1) is a bounded function of the fluctuations on the j+1 level, η (j+1) = η (j+1) δ (j+1) [25][26][27]. Also in general, the harmonic expansion ( 21) is different from the periodic orbit expansion of (19), C As a result, the "integrable spectrum" k (r)…”

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 .…”