Related papers: Numerical evaluation of spherical GJMS determinant…
The conformal anomalies and functional determinants of the Branson--GJMS operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k < d/2+1 (if d is even). The determinants are given in terms…
Motivated by AdS/CFT, the extension is made to spin-half of a scalar calculation of the conformal anomalies and functional determinants of GJMS operators. The formal aspects are heuristic but sufficient. A Barnes zeta function…
GJMS operator determinants in odd dimensions are quickly computed for scalar and spinor fields in both sub- and super-critical cases as sums of Dirichlet eta functions with polynomials in the (integer) operator order as coefficients.
An expression in the form of an easily computed integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a sum formula for the logdet in terms of the logdets of the ordinary…
A numerical expression in the form of an integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a curious sum formula for the logdet in terms of the logdets of the ordinary…
The functional determinant multiplicative anomaly, or defect, is more closely investigated and explicit forms for products of linear operators are produced. I also present formulae for the defect of products of second order operators in…
Two multiplicative anomalies are evaluated for the determinant of the conformal higher spin propagating operator on spheres given by Tseytlin. One holds for the decomposition of the higher derivative product into its individual second order…
It is shown that the functional determinant ($\sim$ effective action) for a scalar field propagating on the mixed signature product of unit spheres, S$^q\times$S$^p$, according to the GJMS operator, depends, if $d$ is odd, only on $d=p+q$…
We compute the scalar determinants $\det(\Delta+M^{2})$ on the two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any finite boundary length $\ell$ and mass $M$, with Dirichlet boundary conditions, using the…
We study a family of higher-derivative conformal operators $P_{2k}^{(2)}$ acting on transverse-traceless symmetric 2-tensors on generic Einstein spaces. They are a natural generalization of the well-known construction for scalars. We first…
In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results to arbitrary…
We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product $H^{\ell} \times S^{d-\ell}$. We also…
We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang…
Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary…
Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a…
For $k \ge 2$ even, and $ \alpha \ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(\alpha,\, \alpha)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
The Dirac APS eta invariant on a Berger sphere of dimension $2n-1$ is discovered, numerically, to coincide, up to spin factors, with the Dirac conformal anomaly on a round sphere of even dimension, $n$. The analytical expression, given in…
In this paper we consider a singular elliptic equation involving the GJMS (Graham-Jenne-Mason-Sparling) operator of order k on n-dimensional compact Riemannian manifold with 2k<n. Mutiplicity and nonexistence results are established.