HST Images of a Galaxy Group at z=2.81, and the Sizes of Damped Lyman Alpha Galaxies

We present HST WFPC2 observations in three bands (F450W, F467M and F814W) of a group of three galaxies at z=2.8 discovered in a ground-based narrow-band search for Lyman alpha emission near the z=2.8 quasar PKS0528-250. One of the galaxies is a damped (DLA) absorber and these observations bear on the relation between the DLA clouds and the Lyman-break galaxies and the stage in the evolution of galaxies they represent. We describe a procedure for combining the undersampled WFPC2 images pointed on a sub-pixel grid, which largely recovers the full sampling of the WFPC2 point spread function (psf). These three galaxies have similar properties to the Lyman-break galaxies except that they have strong Lyman alpha emission. The three galaxies are detected in all three bands, with average B~26, I~25. Two of the galaxies are compact with intrinsic (i.e. after correcting for the effect of the psf) half-light radii of ~0.1 arcsec (~0.4/h kpc, q_o=0.5). The third galaxy comprises two similarly compact components separated by 0.3 arcsec. The HST images and a new ground-based Lyman alpha image of the field provide evidence that the three galaxies are more extended in the light of Lyman alpha than in the continuum. The measured impact parameters for this DLA galaxy (1.17 arcsec), for a second confirmed system, and for several candidates, provide a preliminary estimate of the cross-section-weighted mean radius of the DLA gas clouds at z~3 of less than 13/h kpc, for q_o=0.5. Given the observed sky covering factor of the absorbers this implies that for q_o=0.5 the space density of DLA clouds at these redshifts is more than five times the space density of spiral galaxies locally, with the actual ratio probably considerably greater. For q_o=0.0 there is no evidence as yet that DLA clouds are more common than spiral galaxies locally.Comment: 11 pages, LaTeX, 6 Figures total (4 colour GIF-format, 2 PostScript), accepted for publication in MNRA

Spectral reciprocity and matrix representations of unbounded operators

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. For an infinite discrete set $X$, we consider operators acting on Hilbert spaces of functions on $X$, and their representations as infinite matrices; the focus is on $\ell^2(X)$, and the energy space $\mathcal{H}_{\mathcal E}$. In particular, we prove that these operators are always essentially self-adjoint on $\ell^2(X)$, but may fail to be essentially self-adjoint on $\mathcal{H}_{\mathcal E}$. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the $\mathcal{H}_{\mathcal E}$ operators with the use of a new approximation scheme.Comment: 20 pages, 1 figure. To appear: Journal of Functional Analysi