Refined asymptotic expansion for the partition function of unbounded quantum billiards

Abstract

This paper presents a refined asymptotic expansion for the partition function Θ(t)=Tr e^tΔ of quantum billiards in the unbounded regions {0≤x,0≤yx^μ≤1}, μ>0, and {0≤ye^‖x‖≤1}⊆R², where Δ is the Dirichlet Laplacian. Simon [Ann. Phys. 1 4 6, 209 (1983); J. Funct. Anal. 5 3, 84 (1983)] determined the leading divergence of the trace of the heat kernel for the first class of systems. Standard techniques are combined for the evaluation of Θ for bounded region billiards with results by Van den Berg [J. Funct. Anal. 7 1, 279 (1987)] for ‘‘horn‐shaped regions’’ using an optimized way of dividing the region into ‘‘narrow’’ and ‘‘wide’’ parts to determine the first three terms in the asymptotic expansion of Θ. Results are also stated for bounded regions with cusps that can be obtained by the same method. As an application, the spectral staircase of the strongly chaotic billiard system defined in the region {0≤xy≤2,x≥0}, which has been discussed in connection with the Riemann ζ function and the search for quantum chaos is considered.