Equivalent power law potentials

It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence between power law potentials.Comment: 10 pages. Typographical mistakes in the earlier version are correcte

Tropical rainforest bird community structure in relation to altitude, tree species composition, and null models in the Western Ghats, India

Studies of species distributions on elevational gradients are essential to understand principles of community organisation as well as to conserve species in montane regions. This study examined the patterns of species richness, abundance, composition, range sizes, and distribution of rainforest birds at 14 sites along an elevational gradient (500-1400 m) in the Kalakad-Mundanthurai Tiger Reserve (KMTR) of the Western Ghats, India. In contrast to theoretical expectation, resident bird species richness did not change significantly with elevation although the species composition changed substantially (<10% similarity) between the lowest and highest elevation sites. Constancy in species richness was possibly due to relative constancy in productivity and lack of elevational trends in vegetation structure. Elevational range size of birds, expected to increase with elevation according to Rapoport's rule, was found to show a contrasting inverse U-shaped pattern because species with narrow elevational distributions, including endemics, occurred at both ends of the gradient (below 800 m and above 1,200 m). Bird species composition also did not vary randomly along the gradient as assessed using a hierarchy of null models of community assembly, from completely unconstrained models to ones with species richness and range-size distribution restrictions. Instead, bird community composition was significantly correlated with elevation and tree species composition of sites, indicating the influence of deterministic factors on bird community structure. Conservation of low- and high-elevation areas and maintenance of tree species composition against habitat alteration are important for bird conservation in the southern Western Ghats rainforests.Comment: 36 pages, 5 figures, two tables (including one in the appendix) Submitted to the Journal of the Bombay Natural History Society (JBNHS