Nonperturbative theory of power spectrum in complex systems
Roman Riser, Vladimir Al. Osipov, Eugene Kanzieper
Abstract
The power spectrum analysis of spectral fluctuations in complex wave and quantum systems has emerged as a useful tool for studying their internal dynamics. In this paper, we formulate a nonperturbative theory of the power spectrum for complex systems whose eigenspectra – not necessarily of the random-matrix-theory (RMT) type – possess stationary level spacings. Motivated by potential applications in quantum chaology, we apply our formalism to calculate the power spectrum in a tuned circular ensemble of random N×N unitary matrices. In the limit of infinite-dimensional matrices, the exact solution produces a universal, parameter-free formula for the power spectrum, expressed in terms of a fifth Painlevé transcendent. The prediction is expected to hold universally, at not too low frequencies, for a variety of quantum systems with completely chaotic classical dynamics and broken time-reversal symmetry. On the mathematical side, our study brings forward a conjecture for a double integral identity involving a fifth Painlevé transcendent.