By Klaus Kirsten
The literature at the spectral research of moment order elliptic differential operators features a good deal of data at the spectral services for explicitly identified spectra. a similar isn't real, besides the fact that, for occasions the place the spectra aren't explicitly identified. over the past numerous years, the writer and his colleagues have constructed new, cutting edge equipment for the precise research of various spectral services happening in spectral geometry and less than exterior stipulations in statistical mechanics and quantum box idea. Spectral capabilities in arithmetic and Physics provides a close review of those advances. the writer develops and applies tools for interpreting determinants bobbing up whilst the exterior stipulations originate from the Casimir impact, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta functionality underlies all of those ideas, and the ebook starts off by means of deriving its easy homes and kinfolk to the spectral features. the writer then makes use of these kinfolk to enhance and observe tools for calculating warmth kernel coefficients, practical determinants, and Casimir energies. He additionally explores functions within the non-relativistic context, particularly making use of the options to the Bose-Einstein condensation of a great Bose gas.Self-contained and obviously written, Spectral capabilities in arithmetic and Physics deals a different chance to obtain beneficial new suggestions, use them in various functions, and be encouraged to make additional advances.
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Additional resources for Spectral functions in mathematics and physics
Sample text
2) γ where γ encloses counterclockwise all eigenvalues λk on the real axis. 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 .
Ai (s) = l=0 However, the determinant and the Casimir energy will receive additional contributions from Z(s) and in general an analysis of both these parts is necessary. Up to now we have simply rewritten ζ(s) as N ζ(s) = Z(s) + Ai (s), i=−1 with Z(s) having the properties described. Something has been gained only if the asymptotic terms Ai (s) can be treated analytically in an explicit way. The goal has to be a representation of Ai (s) in terms of known functions and valid in the whole of the complex plane.
Clearly (∂/∂k) ln Jl+1/2 (ka) = aJl+1/2 (ka)/Jl+1/2 (ka) has simple poles at the solutions of eq. 5) with residue 1. So eq. 5) on the positive real axis; see Fig. 1 (for this and a similar treatment of the zeta function as a contour integral see [55, 259, 40, 54]). The above representation of the zeta function is the first step of our procedure. 7) is valid for s > 3/2 only. 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.
