Complex spectral representations and quantum chaos

Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the $k$ -th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the $k$-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.

hep-thCross-listedarXiv:2608.07412Aug 10, 2026

Spectral fluctuations of classically chaotic quantum systems

O. Bohigas, M.J. Giannoni, C. Schmit

Springer eBooksApr 7, 200843 citations

Quantum Mechanics, Diffusion and Chaotic Fractals

Mohamed Saladin El Naschie, Otto E. Rössler, Ilya Prigogine

Quantum mechanics and chaotic fields quantum chaos, complex spectral representation and time-symmetry breaking, T. Petrosky and I. Prigogine scale relativity, fractal space-time and quantum mechanics, L. Nottale introduction to wave phenomena and uncertainty in a fractal space - I, A. Le Mehaute et al young double-slit experiment, Heisenberg uncertainty principle and cantoria space-time, M.S. El Naschie the quantum dimension of space-time, E. Alvarez et al intra-observer chaos - hidden root of quantum mechanics?, O.E. Rossler amplification of superpositional effects through electronic - conformational interactions, M. Conrad quantum cellular automation in 1-D, D.W. Belousek et al an energy-barrier model of biased transport in disordered systems, M. Giona chaos in the Einstein equations, S.E. Rugh.Quantum mechanics and chaotic fields quantum chaos, complex spectral representation and time-symmetry breaking, T. Petrosky and I. Prigogine scale relativity, fractal space-time and quantum mechanics, L. Nottale introduction to wave phenomena and uncertainty in a fractal space - I, A. Le Mehaute et al young double-slit experiment, Heisenberg uncertainty principle and cantoria space-time, M.S. El Naschie the quantum dimension of space-time, E. Alvarez et al intra-observer chaos - hidden root of quantum mechanics?, O.E. Rossler amplification of superpositional effects through electronic - conformational interactions, M. Conrad quantum cellular automation in 1-D, D.W. Belousek et al an energy-barrier model of biased transport in disordered systems, M. Giona chaos in the Einstein equations, S.E. Rugh.

May 1, 1995210 citations

Comment on “new representation of quantum chaos”

Michael Berry

Physics Letters ASep 1, 198410 citations

Quantum chaos: Spectra and states

F. Leyvraz, J. Quezada, T. H. Seligman

Springer eBooksApr 6, 20080 citations

k

-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.