Gaussian transform of the Weil representation

The Pseudo-Character of the Weil Representation and its Relation with the Conley-Zehnder Index

We calculate the character of the Weil representation using previous results which express the Weyl symbol of metaplectic operators in terms of the symplectic Cayley transform and the Conley--Zehnder index.

On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus

We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect to the Weil-Petersson measure. We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian. The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectur…

扭曲拉普拉斯算子的模空间与随机矩阵理论

The moduli space of twisted Laplacians and random matrix theory

本文将 Rudnick 关于双曲曲面拉普拉斯算子谱数方差收敛到高斯正交系综的结果，推广到扭曲拉普拉斯算子收敛到高斯酉系综、Dirac 算子收敛到高斯辛系综，从而回答了 Naud 提出的问题。

Reflections on the q-Fourier transform and the q-Gaussian function

We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be nicely obtained, overcoming all troubles that afflict its real counterpart.

Fourier transform of a Bessel function multiplied by a Gaussian

An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $ I_{mn}(β,q) = \int_{-\infty}^{\infty} e^{-β^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x) \,d x, $ where $m$ and $n$ are non-negative integers, and $J_{n+1/2}(\cdot)$ is a Bessel function of order $n+1/2$.

A simple expression for the four-point scalar function from Gaussian integrals and Fourier transform

Recasting the $N$-point one loop scalar integral from Feynman to Schwinger parameters gives an integrand with a Gaussian form. By application of a Fourier transform, it is easy to derive explicit expressions for the two, three and four-point functions. The Fourier transformation disentangles singularities in the complex plane and extract their contribution as two-point functions in two dimensions. We explicitly derive a one dimensional expression for the (4D) four-point function whose integrand involves only square root and arcsine functions. This report is a condensed version of the approach …