2015/08/09 by P. A. Braun, Petr Braun, Braun, Petr
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Scientific Research and Discoveries #math.DS #nlin.CD #quant-ph
paper · pdf · doi:10.48550/arxiv.1508.02075
arxiv created 2015/08/09 · arxiv updated 2015/08/11
The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the BGS conjecture. We show that the length spectra of their periodic orbits divides into subspectra differing by the parity of the number of reflections from the triangle sides; in the quantum treatment that parity defines the reflection phase of the orbit contribution to the Gutzwiller formula for the energy level density. We apply these results to all 85 arithmetic triangles and establish the boundary conditions under which the quantum billiard is \textquotedblleft genuinely arithmetic\textquotedblright, i. e., has Poissonian level statistics; otherwise the billiard is "pseudo-arithmetic" and belongs to the GOE universality class