119675826
ALGEBRAIC APPROACH TO THE POSITION-DEPENDENT MASS SCHRÖDINGER EQUATION FOR A SINGULAR OSCILLATOR
We construct a singular oscillator Hamiltonian with a position-dependent effective mass. We find that an su(1, 1) algebra is the hidden symmetry of this quantum system and the isospectral potentials V(x) depend on the different choices of the m(x). The complete solutions are also presented by using this Lie algebra.
15491159
Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials
By applying an ansatz to the eigenfunction, an exact closed-form solution of theSchrödinger equation in two dimension is obtained with the potentials V(r) =ar2 + br4 + cr6,V(r) = ar + br2 + cr−1,and V(r) = ar2 + br−2+ cr−4 + dr−6,respectively. The restrictions on the parameters of the given potential andthe angular momentum m are obtained.
119543557
LEVINSON'S THEOREM FOR THE KLEIN-GORDON EQUATION IN TWO DIMENSIONS
The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}\pi=\pi (n_{m}^{+}-n_{m}^{-})= [\delta_{m}(M)+\beta_{1}]-[\delta_{m}(-M)+\beta_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $\delta_{m}$ is named phase shifts. The constants $\beta_{1}$ and $\beta_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.
120777661
ANALYTICAL APPROXIMATIONS TO THE SCHRÖDINGER EQUATION FOR A SECOND PÖSCHL–TELLER-LIKE POTENTIAL WITH CENTRIFUGAL TERM
The bound state solutions of the Schrodinger equation for a second Poschl–Teller-like potential with the centrifugal term are obtained approximately. It is found that the solutions can be expressed in terms of the hypergeometric functions 2F1(a, b; c; z). To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other method for short-range potential. Two special cases for l = 0 and V1 = V2 are also studied briefly.
117217126
Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz
Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+λ), the singular one-fraction power one V(r)=ar−1/2+br−3/2 and the singular two-fraction one V(r)=ar2/3+br−2/3+cr−4/3. The latter one is found that the hidden symmetry exists if substituting r→−ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.
121783508
Quantum states of a sextic potential: hidden symmetry and quantum monodromy
A known terminating polynomial ansatz for a finite sub-set of states of the sextic central potential V(a,b,c;r) = ar2 + br4 + cr6, c>0, subject to integer related parameter constraints, is confirmed to apply in two as well as three dimensions. A hidden symmetry relating the eigenstates of V(a,b,c;r) to those of V(a,-b,c;r) is used to establish the boundary between the ansatz sub-set and the infinite set of higher states. Connections between the radial wavefunctions Rn(a,b,c;r) and Rn'(a,-b,c;r) also provide semiclassical insight into the origin of the parameter constraint. Finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy.
40006839
Water in HD 209458b's atmosphere from 3.6 - 8 microns IRAC photometric observations in primary transit
The hot Jupiter HD 209458b was observed during primary transit at 3.6, 4.5, 5.8 and 8.0 microns using the Infrared Array Camera (IRAC) on the Spitzer Space Telescope. We detail here the procedures we adopted to correct for the systematic trends present in the IRAC data. The light curves were fitted including limb darkening effects and fitted using Markov Chain Monte Carlo and prayer-bead Monte Carlo techniques, finding almost identical results. The final depth measurements obtained by a combined Markov Chain Monte Carlo fit are at 3.6 microns, 1.469 +- 0.013 % and 1.448 +- 0.013 %; at 4.5 microns, 1.478 +- 0.017 % ; at 5.8 microns, 1.549 +- 0.015 % and at 8.0 microns 1.535 +- 0.011 %. Our results clearly indicate the presence of water in the planetary atmosphere. Our broad band photometric measurements with IRAC prevent us from determining the additional presence of other other molecules such as CO, CO2 and methane for which spectroscopy is needed. While water vapour with a mixing ratio of 10^-4-10^-3 combined with thermal profiles retrieved from the day-side may provide a very good fit to our observations, this data set alone is unable to resolve completely the degeneracy between water abundance and atmospheric thermal profile.
118316787
An exact integral relation between the Ni56 mass and the bolometric light curve of a type Ia supernova
An exact relation between the Ni56 mass and the bolometric light curve of a type Ia supernova can be derived as follows, using the following excellent approximations: 1. the emission is powered solely by Ni56-> Co56 ->Fe56; 2. each mass element propagates at a non-relativistic velocity which is constant in time (free coasting); and 3. the internal energy is dominated by radiation. Under these approximations, the energy E(t) carried by radiation in the ejecta satisfies: dE/dt=-E(t)/t-L(t)+Q(t), where Q(t) is the deposition of energy by the decay which is precisely known and L(t) is the bolometric luminosity. By multiplying this equation by time and integrating over time we find: E(t)*t=\int_0^t Q(t')t'dt' -\int_0^t L(t')t'dt'. At late time, t>> t_peak, the energy inside the ejecta decreases rapidly due to its escape, and thus we have \int_0^t Q(t')t'dt'=\int_0^t L(t')t'dt'. This relation is correct regardless of the opacities, density distribution or Ni56 deposition distribution in the ejecta and is very different from "Arnett's rule", L_peak ~ Q(t_peak). By comparing \int_0^t Q(t')t'dt' with \int_0^t L(t')t'dt' at t~40 day after the explosion, the mass of Ni56 can be found directly from UV, optical and infrared observations with modest corrections due to the unobserved gamma-rays and due to the small residual energy in the ejecta, E(t)*t>0.
119233296
THE STATISTICS OF MULTI-PLANET SYSTEMS
We describe statistical methods for measuring the exoplanet multiplicity function (the fraction of host stars containing a given number of planets) and inclination distribution from transit and radial-velocity (RV) surveys. The analysis is based on the approximation of separability-that the distribution of planetary parameters in an n-planet system is the product of identical 1-planet distributions. We review the evidence that separability is a valid approximation for exoplanets and conclude that it captures many, but not all, of the known characteristics of multi-planet systems. We show how to relate the observable multiplicity function in surveys with similar host-star populations but different sensitivities. We also show how to correct for geometrical selection effects to derive the multiplicity function from transit surveys if the distribution of relative inclinations is known. Applying these tools to the Kepler transit survey and to RV surveys, we find that (1) the Kepler data alone do not constrain the mean inclination of multi-planet systems; even spherical distributions are allowed by the data but only if a small fraction of host stars contain large planet populations ({approx}> 30); (2) comparing the Kepler and RV surveys shows that the mean inclination of multi-planet systems is less than 5 Degree-Sign ;more » and (3) the multiplicity function of the Kepler planets is not well determined by the present data.« less
7872671
The Demographics of Extrasolar Planets Beyond the Snow Line with Ground-based Microlensing Surveys
In the currently-favored paradigm of planet formation, the location of the snow line in the protoplanetary disk plays a crucial role. Determining the demographics of planets beyond the snow line of stars of various masses is thus essential for testing this model. Microlensing is sensitive to planets that are generally inaccessible to other methods, and in particular is most sensitive to cool planets at or beyond the snow line, including very low-mass (i.e. terrestrial) planets. Hence, microlensing is uniquely suited and so essential for a comprehensive study of this region. Microlensing is also sensitive to planets orbiting low-mass stars, free-floating planets, planets in the Galactic bulge and disk, and even planets in external galaxies. These planets can also provide critical constraints on models of planet formation. Although microlensing searches have so far detected only a handful of planets, these have already changed our understanding of planet formation beyond the snow line. Next generation microlensing surveys, which would be sensitive to tens of "cold Earths" in this region, are well advanced in design conception and are starting initial practical implementation.
119112522
Type Ia supernovae with bimodal explosions are common – possible smoking gun for direct collisions of white dwarfs
We discover clear doubly-peaked line profiles in 3 out of ~20 type Ia supernovae (SNe Ia) with high-quality nebular-phase spectra. The profiles are consistently present in three well-separated Co/Fe emission features. The two peaks are respectively blue-shifted and red-shifted relative to the host galaxies and are separated by ~5000 km/s. The doubly-peaked profiles directly reflect a bi-modal velocity distribution of the radioactive Ni56 in the ejecta that powers the emission of these SNe. Due to their random orientations, only a fraction of SNe with intrinsically bi-modal velocity distributions will appear as doubly-peaked spectra. Therefore SNe with intrinsic bi-modality are likely common, especially among the SNe in the low-luminosity part on the Philips relation (\Delta m15(B) >~ 1.3; ~40% of all SNe Ia). Such bi-modality is naturally expected from direct collisions of white dwarfs (WDs) due to the detonation of both WDs and is demonstrated in a 3D 0.64 M_Sun-0.64 M_Sun WD collision simulation. In the future, with a large sample of nebular spectra and a comprehensive set of numerical simulations, the collision model can be unambiguously tested as the primary channel for type Ia SNe, and the distribution of nebular line profiles will either be a smoking gun or rule it out.