2009/02/12 by Wu-Sheng Dai, Mi Xie
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Quantum Mechanics and Applications #Statistical Mechanics and Entropy #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1126-6708/2009/02/033
published as JHEP02(2009)033 · 17 pages, 1 figure. v2: Equivalent forms of Eqs. (4.8) and (9.2) are added
openalex publication_date 2009/02/12 · arxiv created 2009/03/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.