1996/02/22 by Francesco Mezzadri, Mezzadri, Francesco, Scotti, Antonio
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.chao-dyn/9602021
openalex publication_date 1996/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show that the quantum theory of chaos, based on the statistical theory of energy spectra, presents inconsistencies difficult to overcome. In classical mechanics a system described by an hamiltonian H = H1 + H2 (decomposable) cannot be ergodic, because there are always two dependent integrals of motion besides the constant of energy. In quantum mechanics we prove the existence of decomposable systems \linebreak Hq = Hq1 + Hq2 whose spacing distribution agrees with the Wigner law and we show that in general the spacing distribution of Hq is not the Poisson law, even if it has often the same qualitative behaviour. We have found that the spacings of Hq are among the solutions of a well defined class of homogeneous linear systems. We have obtained an explicit formula for the bases of the kernels of these systems, and a chain of inequalities which the coefficients of a generic linear combination of the basis vectors must satisfy so that the elements of a particular solution will be all positive, i.e. can be considered a set of spacings.