7,081 research outputs found
Heat-kernel expansion on non compact domains and a generalised zeta-function regularisation procedure
Heat-kernel expansion and zeta function regularisation are discussed for
Laplace type operators with discrete spectrum in non compact domains. Since a
general theory is lacking, the heat-kernel expansion is investigated by means
of several examples. It is pointed out that for a class of exponential
(analytic) interactions, generically the non-compactness of the domain gives
rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic
continuation of the associated zeta function is investigated. A simple model is
considered, for which the analytic continuation of the zeta function is not
regular at the origin, displaying a pole of higher order. For a physically
meaningful evaluation of the related functional determinant, a generalised zeta
function regularisation procedure is proposed.Comment: Latex, 14 pages, no figures. The version to be published in JM
The Lerch Zeta Function II. Analytic Continuation
This is the second of four papers that study algebraic and analytic
structures associated with the Lerch zeta function. In this paper we
analytically continue it as a function of three complex variables. We that it
is well defined as a multivalued function on the manifold M equal to C^3 with
the hyperplanes corresponding to integer values of the two variables a and c
removed. We show that it becomes single valued on the maximal abelian cover of
M. We compute the monodromy functions describing the multivalued nature of this
function on M, and determine various of their properties.Comment: 29 pages, 3 figures; v2 notation changes, homotopy action on lef
Period functions for Maass wave forms. I
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a
smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to
C which is small as y \to \infty and satisfies Delta u = lambda u for some
lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic
Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H,
with the usual trigonometric waveforms on the torus R^2/Z^2, which are also
(for this surface) both the Fourier building blocks for L_2 and eigenfunctions
of the Laplacian. Although therefore very basic objects, Maass forms
nevertheless still remain mysteriously elusive fifty years after their
discovery; in particular, no explicit construction exists for any of these
functions for the full modular group. The basic information about them (e.g.
their existence and the density of the eigenvalues) comes mostly from the
Selberg trace formula: the rest is conjectural with support from extensive
numerical computations.Comment: 68 pages, published versio
Large-order Perturbation Theory and de Sitter/Anti de Sitter Effective Actions
We analyze the large-order behavior of the perturbative weak-field expansion
of the effective Lagrangian density of a massive scalar in de Sitter and anti
de Sitter space, and show that this perturbative information is not sufficient
to describe the non-perturbative behavior of these theories, in contrast to the
analogous situation for the Euler-Heisenberg effective Lagrangian density for
charged scalars in constant electric and magnetic background fields. For
example, in even dimensional de Sitter space there is particle production, but
the effective Lagrangian density is nevertheless real, even though its
weak-field expansion is a divergent non-alternating series whose formal
imaginary part corresponds to the correct particle production rate. This
apparent puzzle is resolved by considering the full non-perturbative structure
of the relevant Feynman propagators, and cannot be resolved solely from the
perturbative expansion.Comment: 18 page
- …