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 $E=\pm M$.
8332580
The Exact Solution to the Schr\"{o}dinger Equation with the Octic Potential
The Schr\"{o}dinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schr\"{o}dinger equation for the radial wave function through a simple transformation. As an example, applying an ${\it ansatz}$ to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schr\"{o}dinger equation with the octic potential, $V(r)=ar^2-br^4+cr^6-dr^4+er^{10}$.
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.
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.
119219662
The rate of WD-WD head-on collisions may be as high as the SNe Ia rate
We show that a White Dwarf-White Dwarf (WD-WD) binary with semi-major axis a=1-300 AU, which is orbited by a stellar mass outer perturber with a moderate pericenter r_{p, out} \sim 3-10 x a, has a few percent chance of experiencing a head-on collision within ~5 Gyr. Such a perturber is sufficiently distant to allow the triple system to remain intact for millions of orbits while efficiently exchanging angular momentum with the WD-WD binary. In ~ 5% of the initial orientations, the inner orbit efficiently scans the (equal energy) phase space in the region of zero angular momentum. In these systems, the binary experiences increasingly closer, stochastic, pericenter approaches r_p ~ a/2N with the increasing number (N) of orbits elapsed. Within N~10^5(a/30AU) orbits, a collision is likely to occur. This is shown by performing \simten thousand 3-body integrations and is explained by simple analytic arguments. The collisions are conservatively restricted to "clean" collisions in which all passages prior to the collision are greater than 4R_WD=4x10^9cm. In particular, within the last single orbit, the pericenter changes from r_p>4R_WD to a collision value of r_p 3000 km/s and the collision is likely to detonate the WDs leading to a type Ia SNe. If a significant fraction of WDs reside in such triples, the rate of such collisions is as high as the SNe Ia rate, and it is possible that some or all type Ia SNe occur in this way. Such SNe have a unique gravitational wave signature, which will allow a decisive identification in the future.
119217995
SUB-SATURN PLANET MOA-2008-BLG-310Lb: LIKELY TO BE IN THE GALACTIC BULGE*
We report the detection of sub-Saturn-mass planet MOA-2008-BLG-310Lb and argue that it is the strongest candidate yet for a bulge planet. Deviations from the single-lens fit are smoothed out by finite-source effects and therefore are not immediately apparent from the light curve. Nevertheless, we find that a model in which the primary has a planetary companion is favored over the single-lens model by Δχ2 ~ 880 for an additional 3 degrees of freedom. Detailed analysis yields a planet/star mass ratio q = (3.3 ± 0.3) × 10–4 and an angular separation between the planet and star within 10% of the angular Einstein radius. The small angular Einstein radius, θ E = 0.155 ± 0.011 mas, constrains the distance to the lens to be DL >6.0 kpc if it is a star (ML >0.08 M ). This is the only microlensing exoplanet host discovered so far that must be in the bulge if it is a star. By analyzing VLT NACO adaptive optics images taken near the baseline of the event, we detect additional blended light that is aligned to within 130 mas of the lensed source. This light is plausibly from the lens, but could also be due to a companion to the lens or source, or possibly an unassociated star. If the blended light is indeed due to the lens, we can estimate the mass of the lens, ML = 0.67 ± 0.14 M , planet mass m = 74 ± 17 M ⊕, and projected separation between the planet and host, 1.25 ± 0.10 AU, putting it right on the "snow line." If not, then the planet has lower mass, is closer to its host and is colder. To distinguish among these possibilities on reasonable timescales would require obtaining Hubble Space Telescope images almost immediately, before the source-lens relative motion of U=5 mas yr -1 causes them to separate substantially.
118852244
Exponential growth of eccentricity in secular theory
The Kozai mechanism for exponentially exciting eccentricity of a Keplerian orbit by a distant perturber is extended to a general perturbing potential. In particular, the case of an axisymmetric potential is solved analytically. The analysis is applied to orbits around an oblate central object with a distant perturber. If the equatorial plane of the central object is aligned with the orbit of the distant perturber (axisymmetric potential), a single instability zone, in which eccentricity grows exponentially, is found between two critical inclinations; if misaligned (non-axisymmetric potential), a rich set of critical inclinations separating stable and unstable zones is obtained (Vashkoviak 1974). The analysis is also applied to a general quadratic potential. Similarly, for non-axisymmetric cases, multiple stability and instability zones are obtained. Here eccentricity can reach very high values in the instability zones even when the potential's deviation from axisymmetry is small.
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.