WKB Approximation to the Power Wall

We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^{\alpha}$ for $x>0$; the limit $\alpha\to\infty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $\alpha =1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.Comment: 30 pages, 8 figures. Minor corrections in v.

The Galactic disk mass function: reconciliation of the HST and nearby determinations

We derive and parametrize the Galactic mass function (MF) below 1 \msol characteristic of both single objects and binary systems. We resolve the long standing discrepancy between the MFs derived from the HST and from the nearby luminosity functions, respectively. We show that this discrepancy stemmed from {\it two} cumulative effects, namely (i) incorrect color-magnitude determined distances, due a substantial fraction of M dwarfs in the HST sample belonging to the metal-depleted, thick-disk population, as corrected recently by Zheng et al. and (ii) unresolved binaries. We show that both the nearby and HST MF for unresolved systems are consistent with a fraction $\sim$50% of M-dwarf binaries, with the mass of both the primaries and the companions originating from the same underlying single MF. This implies that $\sim$30% of M dwarfs should have an M dwarf companion and $\sim$20% should have a brown dwarf companion, in agreement with recent determinations. The present calculations show that the so-called "brown-dwarf desert" should be reinterpreted as a lack of high mass-ratio (m_2/m_1\la 0.1) systems, and does not preclude a substantial fraction of brown dwarfs as companions of M dwarfs or for other brown dwarfs.Comment: 16 pages, Latex file, uses aasms4.sty, to appear in ApJ Letter

Asymptotic self-consistency in quantum transport calculations

Ab initio simulations of quantum transport commonly focus on a central region which is considered to be connected to infinite leads through which the current flows. The electronic structure of these distant leads is normally obtained from an equilibrium calculation, ignoring the self-consistent response of the leads to the current. We examine the consequences of this, and show that the electrostatic potential Delta phi is effectively being approximated by the difference between electrochemical potentials Delta mu, and that this approximation is incompatible with asymptotic charge neutrality. In a test calculation for a simple metal-vacuum-metal junction, we find significant errors in the nonequilibrium properties calculated with this approximation, in the limit of small vacuum gaps. We provide a scheme by which these errors may be corrected