PhD thesis
In this work we determine the
wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space.
Let $G = SO_{1,n}{(R)}_e$ be the connected component of identity of Lorentz group and let $H = SO_{1,n-1}{(R)}_e$, a subset of G.
The
de Sitter space $dS^n$, is the one-sheeted hyperboloid in $R^{1,n}$ isomorphic to G/H.
A spherical distribution, is an H-invariant, eigendistribution of the Laplace-Beltrami operator on $dS^n$. T
he space of spherical distributions with eigenvalue $\lambda$, denoted by $D'_{\lambda}(dS^n)$, has dimension 2.
We construct a basis for the space of positive-definite spherical distributions as boundary value of
sesquiholomorphic kernels on the crown domains, an open G-invariant domain in $dS^n_C$. It contains $dS^n$ as a G-orbit on the boundary.
We characterize the analytic wavefront set for such distributions. Moreover, if a spherical distribution $\Theta$ in $D'_{\lambda}(dS^n)$ has the wavefront set same as one
of the basis element, then it must be a constant multiple of that basis element. Using the analytic wavefront sets we show that the basis elements
of $D'_{\lambda}(dS^n)$ can not vanish in any open region.
Research Publications
- G. Olafsson, I.Sitiraju: "Analytic Wavefront Sets of Spherical Distributions on the De Sitter space". arXiv:2309.10685
Master's thesis
The aim of my thesis was to understand and analyze whether there are uniformly
discrete sets other than lattices which are generated by standard basis of R^n (i.e. Z^n)
for which the Poisson summation formula holds in the sense of tempered distribution.
Click
here to my master's thesis.