The Local Geometric Asymptotics of Continuum Eigenfunction Expansions I. Overview

Abstract

The well-known connection between (a) the asymptotic density of eigenvalues of a differential operator H, and (b) the geometry of the region or manifold where H acts, has a local generalization: There is a connection between (a') the spectral measures or projection kernel describing the proper normalization, relative to a point x 0, of the expansion of an arbitrary function in eigenfunctions of an operator H (possibly with continuous spectrum), and (b') the values of the coefficients (symbol) of H and their derivatives at x 0.Potential applications (especially in general-relativistic quantum field theory) increasingly call for a detailed development of this theory (including calculation of numerical coefficients), which is begun here. The small-time expansion of the Green function of the heat operator, + H, is used to define the "mean" or "effective" expansion of the spectral measures. For the case of one independent and one dependent variable, the spectral and heat expansions are worked out in detail to a rather high order. They are shown to be related to (and obtainable from) existing phase-integral expansions with local amplitude functions (WKB-FrSman expansions) of the eigenfunctions.

Vacuum Energy as Spectral Geometry

“…First, integrate σ to get a local analogue of the counting function. (This is the inverse Laplace transform of the diagonal value of the heat kernel; it is the quantity called µ 00 in [25].) As expected,…”

“…The inverse Laplace transform of the leading term in Tr K gives the leading behavior at large λ of the density of eigenvalues, and the higher-order terms correspond similarly to lower-order corrections to the eigenvalue distribution, on the average [12,40,25]. (When V (x) is a confining potential [36,2,9], H = −∇ 2 + V may have discrete spectrum even though its spatial domain, Ω, is not compact.…”

Some subtleties in the relationships among heat kernel invariants, eigenvalue distributions, and quantum vacuum energy

“…Because Tr K is the Laplace transform of dN dλ , one can show, by calculations like those in [5], that the coefficients in this series, if it existed, would be determined by the coefficients a s [Ω] in the rigorous asymptotic series (4). (The Laplace transform of λ p−1 is proportional to t −p , at least for p > 0.)…”

Spectral asymptotics in eigenvalue problems with nonlinear dependence on the spectral parameter

“…The reason is that starting with certain n these terms become smaller then fluctuations of N(λ, ω) when λ goes from one eigen-value to the next one [6]. The way out of this difficulty is to work with smoothed functions N(λ, ω) and ϕ(λ, ω), see, e.g., [6], [7]. The smoothing can be done by different ways and one of them is to use the Riesz means [8].…”