By Klaus Kirsten
The literature at the spectral research of moment order elliptic differential operators incorporates a good deal of knowledge at the spectral services for explicitly recognized spectra. an identical isn't actual, besides the fact that, for events the place the spectra are usually not explicitly identified. during the last a number of years, the writer and his colleagues have constructed new, leading edge equipment for the precise research of various spectral services taking place in spectral geometry and below exterior stipulations in statistical mechanics and quantum box idea. Spectral services in arithmetic and Physics provides an in depth evaluate of those advances. the writer develops and applies tools for interpreting determinants coming up whilst 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 publication starts off by means of deriving its easy homes and relatives to the spectral services. the writer then makes use of these family to advance and practice tools for calculating warmth kernel coefficients, sensible determinants, and Casimir energies. He additionally explores purposes within the non-relativistic context, particularly utilizing the ideas to the Bose-Einstein condensation of a fantastic Bose gas.Self-contained and obviously written, Spectral features in arithmetic and Physics bargains a distinct chance to procure invaluable new innovations, use them in various functions, and be encouraged to make additional advances.
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Additional info for Spectral Functions in Mathematics and Physics
Given that the interesting properties of the zeta function (namely nearly all heat kernel coefficients of (−∆), the determinant and the Casimir energy) are encoded to the left of the strip 1/2 < s < 1, how can we find the analytical continuation of it to this range? As explained, the restriction 1/2 < s is a result of the behaviour of the integrand as k → ∞. 9), the strip of convergence will certainly move to the left. So if the asymptotic terms alone can be treated analytically and ζ ν (s) with the asymptotic terms subtracted can be dealt with at least numerically, the analytic continuation can be found.
However, before proceeding to these different applications we will discuss general aspects of the described procedure. 2 Scalar field on the D-dimensional generalized cone For the applications to come, it will be very important that results in arbitrary dimensions are available. Furthermore, as we will see, no additional complication will arise by not considering the ball as the underlying manifold, but instead what can be termed the bounded generalized cone. The relevant approach has been developed by Bordag, Dowker and Kirsten .
9), there will be imaginary zeroes for S > 1+ν −D/2 and for convenience we restrict attention to S ≤ 1 + ν − D/2. This will be sufficient for our purposes. This concludes our consideration of a general base manifold N because ¯ 2 the base zeta function ζN cannot be analysed without any knowledge of λ further. So let us now see how far we can go when specializing to a “simple” base manifold, where simple might be defined to mean cases for which the eigenvalues λ are known. Examples that come immediately to mind are the sphere or the torus.