Scarring Effects on Tunneling in Chaotic Double-Well Potentials

The connection between scarring and tunneling in chaotic double-well potentials is studied in detail through the distribution of level splittings. The mean level splitting is found to have oscillations as a function of energy, as expected if scarring plays a role in determining the size of the splittings, and the spacing between peaks is observed to be periodic of period {$2\pi\hbar$} in action. Moreover, the size of the oscillations is directly correlated with the strength of scarring. These results are interpreted within the theoretical framework of Creagh and Whelan. The semiclassical limit and finite-{$\hbar$} effects are discussed, and connections are made with reaction rates and resonance widths in metastable wells.Comment: 22 pages, including 11 figure

Multidimensional extension of the generalized Chowla-Selberg formula

After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form \zeta_{A,\vec{b},q} (s) = \sum_{\vec{n}\in Z^p (\vec{n}^T A \vec{n} +\vec{b}^T \vec{n}+q)^{-s}, with $A$ the $p\times p$ matrix of a quadratic form, $\vec{b}$ a $p$ vector and $q$ a constant, is obtained. It is valid on the whole complex $s$-plane, is exponentially convergent and provides the residua at the poles explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case $p=2$, $\vec{b}= \vec{0}$, $q=0$. Some variations of the formula and physical applications are considered.Comment: LaTeX, 15 pages, no figure

Zeta function regularization in Casimir effect calculations and J.S. Dowker's contribution

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.Comment: 16 pages, dedicated to J.S. Dowker, version to appear in International Journal of Modern Physics

Integration over connections in the discretized gravitational functional integrals

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas) i. e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular ($\delta$-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some another features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.Comment: 22 page