The number of eigenstates: counting function and heat kernel

Abstract

The main aim of this paper is twofold: (1) revealing a relation between the counting function N (λ) (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 (λ), for there is no effective method for calculating N (λ) beyond leading order. We suggest a new expression of N (λ) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N (λ) from a nonuniformly convergent expansion of K (t). We calculate N (λ) 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 (λ) are introduced.

An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions

“…[9], in which the renormalization procedure is based on the analytical continuation of the gamma function.…”

“…From a counting function, one can immediately achieve the eigenvalue spectrum of the operator D [9]. By N (λ n ) = n, we can obtain the spectrum, Then for H 3 and its quotient H 3 /Z, we have 12) where y = (ξ, η) and γy = γ (ξ, η) → |λ| −1 ξ, λ −1 η with λ = e i2πτ and τ = (θ + iβ) /2π;…”

“…Moreover, starting from a counting function, we calculate the eigenvalue spectrum of the operator D using the approach given in Ref. [9].…”

Deriving time-dependent diffusion and relaxation rate in porous systems using eigenfunctions of the Laplace operator

“…Many open questions remain regarding the connection between the trace function and the density function, but a recent result treats the above result in terms of the density function [23] NðkÞ…”

Anisotropic Pressure of a Quantum Gas in Confined Space

“…19 and 20, the authors studied the mathematical basis concerning the boundary effects, while in Ref. 21 the authors discussed the statistical thermodynamics for a non-extensive system generally.…”