By Andrei A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, S. Zerbini

One of many goals of this booklet is to provide an explanation for in a uncomplicated demeanour the possible tricky problems with mathematical constitution utilizing a few particular examples as a consultant. In all of the circumstances thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly validated. The authors attempt to fill the space that usually exists among the physics of quantum box theories and the mathematical tools most fitted for its formula, that are more and more hard at the mathematical skill of the physicist.

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**Example text**

In the first case, A is the Euclidean operator of motion of charged bosons, in the latter A is the Euclidean operator of motion of neutral bosons. A few comments on these hypotheses are in order. First of all, a countable base of the topology is required in order to endow the manifold with a par tition of the unity and allow the use of Hilbert-Schmidt's theory (which is Survey of the Chapter, Notation and Conventions 31 fundamental in our dealing with the heat-kernel theory [I. Chavel (1984)]).

54), one immediately gets (here S[

A(n) = — (dT — M) 2 + LN and the latter equation defines the thermodynamic poten tial £1(13, fj). The operator A((i) is still elliptic but non-Hermitian, in fact Static Spacetimes: Thermodynamic Effects 19 it is normal and its eigenvalues are complex and read ( 27m \2 — + i/j,J +co], n = 0,±l,±2,... 59) Nevertheless, one can still define the related zeta function. , fi) = -pS[4>c,g] - ±C'(0\PA(ri) . 60) Using Eq. l) in Appendix A and the Mellin representation for ((s\A), Eq. A. Bytsenko et al.