Characterization of almost LpL^p-eigenfunctions of the Laplace-Beltrami operator

In \cite{Roe} Roe proved that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R$, then $f_0(x)=a\sin(x+\varphi)$ where $a$ and $\varphi$ are real constants. This result was extended to $\R^n$ by Strichartz \cite{Str} where $d/dx$ is substituted by the Laplacian on $\R^n$. While it is plausible to extend this theorem for other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic 3-space. This negative result can be indeed extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the $p$-dependance of the $L^p$-spectrum of the Laplacian on the hyperbolic spaces. Taking this into account we shall prove that for all rank one Riemannian symmetric spaces of noncompact type, or more generally for the harmonic $NA$ groups, the theorem actually holds true when uniform boundedness is replaced by uniform "almost $L^p$ boundedness". In addition we shall see that for the symmetric spaces this theorem is capable of characterizing the Poisson transforms of $L^p$ functions on the boundary, which some what resembles the original theorem of Roe on $\R$.Comment: 30 page

Anisotropic cosmological models with two fluids

In this paper, aniostropic dark energy cosmological models have been constructed in a Bianchi-V space-time with the energy momentum tensor consisting of two non-interacting fluids namely bulk viscous fluid and dark energy fluid. Two different models are constructed based on the power law cosmology and de Sitter universe. The constructed model also embedded with different pressure gradients along different spatial directions. The variable equation of state (EoS) parameter, skewness parameters for both the models are obtained and analyzed. The physical properties of the models obtained with the use of scale factors of power law and de Sitter law are also presented.Comment: 10 pages, 12 figure

Possible existence of wormholes in the galactic halo region

Two observational results, the density profile from simulations performed in the $\Lambda$CDM scenario and the observed flat galactic rotation curves, are taken as input with the aim of showing that the galactic halo possesses some of the characteristics needed to support traversable wormholes. This result should be sufficient to provide an incentive for scientists to seek observational evidence for wormholes in the galactic halo region.Comment: 11 pages, 3 figures, 1 table, Eur. Phys. J. C (2014) 74:2750, DOI 10.1140/epjc/s10052-014-2750-