By Klaus Kirsten
ISBN-10: 1420035460
ISBN-13: 9781420035469
ISBN-10: 158488259X
ISBN-13: 9781584882596
The literature at the spectral research of moment order elliptic differential operators includes a good deal of data at the spectral features for explicitly identified spectra. a similar isn't real, even if, for events the place the spectra are usually not explicitly identified. during the last numerous years, the writer and his colleagues have built new, cutting edge tools for the precise research of numerous spectral features taking place in spectral geometry and below exterior stipulations in statistical mechanics and quantum box thought. Spectral capabilities in arithmetic and Physics provides an in depth evaluate of those advances. the writer develops and applies equipment for reading determinants bobbing up whilst the exterior stipulations originate from the Casimir impression, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta functionality underlies all of those ideas, and the publication starts by means of deriving its easy homes and relatives to the spectral services. the writer then makes use of these kinfolk to advance and follow equipment for calculating warmth kernel coefficients, useful determinants, and Casimir energies. He additionally explores purposes within the non-relativistic context, particularly utilizing the strategies to the Bose-Einstein condensation of an incredible Bose gas.Self-contained and obviously written, Spectral services in arithmetic and Physics deals a distinct chance to procure helpful new concepts, use them in quite a few functions, and be encouraged to make additional advances.
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Additional resources for Spectral functions in mathematics and physics
Example text
15) These results can be read from eqs. 5) for ν 2 is used and 2ζH (2s − 1; 1/2) is replaced by the base zeta function. Considering the asymptotics of the integrand in eq. 12) and by having in mind that the ν 2 are eigenvalues of a second-order differential operator, see eqs. 5), the function Z(s) is seen to be analytic on the strip (d − 1 − N )/2 < s. As is clearly apparent in eqs. 15), base contributions are separated from radial ones. 1) is considered is completely encoded in the numerical multipliers xi,b .
Here, the Green’s function Gλ (x, x ) is the kernel of Rλ , Rλ f (x) = dx Gλ (x, x )f (x ). 4) in terms of a complete set of normalized eigenfunctions φk . 4) is merely formal. 4). Instead, pseudo-differential calculus provides an effective tool to find precisely this information. Instead of dealing with the differential operators themselves, we work with their symbols in Fourier space. First note that K(t, x, x ), eq. 2), is the kernel of the operator e−tP = i 2π dλ e−λt (P − λ)−1 . 5) γ We want to find the resolvent Rλ defined by (P − λ)Rλ = 1, or at least, for the reason mentioned, a large-|λ| approximation.
27) is a local condition, the crucial difference is that a projection onto the space spanned by the eigenfunctions ϕj (w) with positive, respectively, negative eigenvalues is involved. , 1 A + |A| , 2 |A| Π> = so Π> is a pseudo-differential operator of order 0 and we leave the class of situations considered previously. 27). Given our choice of example, we do not need to resort to a symbol calculus. Instead, we express the heat trace and the zeta function of P in terms of these quantities for A2 .