By Klaus Kirsten
The literature at the spectral research of moment order elliptic differential operators encompasses a good deal of knowledge at the spectral capabilities for explicitly recognized spectra. a similar isn't actual, notwithstanding, for events the place the spectra usually are not explicitly identified. over the past a number of years, the writer and his colleagues have constructed new, cutting edge equipment for the precise research of a number of spectral features taking place in spectral geometry and lower than exterior stipulations in statistical mechanics and quantum box idea. Spectral features in arithmetic and Physics offers an in depth review of those advances. the writer develops and applies tools for reading determinants coming up while the exterior stipulations originate from the Casimir impact, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta functionality underlies all of those options, and the e-book starts through deriving its simple houses and family members to the spectral features. the writer then makes use of these kinfolk to increase and observe equipment for calculating warmth kernel coefficients, practical determinants, and Casimir energies. He additionally explores functions within the non-relativistic context, particularly employing the concepts to the Bose-Einstein condensation of an awesome Bose gas.Self-contained and obviously written, Spectral capabilities in arithmetic and Physics deals a different chance to procure precious new suggestions, use them in quite a few purposes, and be encouraged to make additional advances.
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For reasons explained in Chapter 2, we are especially interested in the properties of ζ(s) in the range s < 3/2 and therefore we need to perform the analytical continuation to the left. Leaving out the l-summation for the moment, we will first proceed with the k-integral alone. ©2002 CRC Press LLC The first specific idea is to shift the integration contour and place it along the imaginary axis. If we look at the origin, we see that for k → 0, to leading order, we have the behavior Jν (k) ∼ k ν /(2ν Γ(ν + 1)) such that the integrand diverges at this limit.
17) at r = 0 only. Defining the Fourier transform with respect to y, 1 f˜(ω, s) = dy e−iωy f (y, s), (2π)D/2 IRD−1 the Fourier transform fˆ(ω, τ ) is rewritten as ∞ 1 f (ω, τ ) = √ 2π ds e−isτ f˜(ω, s), −∞ and H−2−j is cast into the form ∞ H−2−j f (y, r) = 1 ˜ −2−j (y, r, ω, τ, λ)f˜(ω, s). , Γ− = (∞, −∞). Note the homogeneity property ω λ ˜ −2−j y, ρ, √ , σ, h t t =t 1+j 2 σ ˜ −2−j y, √ρ , ω, √ ,λ . h t t For the kernel of H−2−j , on the diagonal, this shows H−2−j (y, r, y, r, λ) = 1 (2π)D ˜ −2−j (y, r, ω, r, λ).
In this case the base zeta function is ∞ (2l + d − 1) ζN (s) = l=0 (l + d − 2)! (d − 1)! l+ −2s d−1 2 . The degeneracy can be written as d(l) = l+d−1 d−1 + l+d−2 d−1 , and it is seen immediately that ζN (s) is a sum of Barnes zeta functions [35, 34] defined as ∞ ζB (s, b) = m=0 ∞ = l=0 1 (b + m1 + ... + md )s l+d−1 d−1 (l + b)−s . ) In detail we find the relation ζN (s) = ζB 2s, d+1 2 + ζB 2s, d−1 2 . 21) A slightly more complicated situation is if N is not a unit d-sphere but a sphere of radius a.