119892879
Schrödinger Equation with a Coulomb Field in 2+1 Dimensions
The solutions of the two-dimensional Schrodinger equation with a Coulomb potential are briefly studied and compared with those of the Klein-Gordon case studied in this work. The eigenvalues and the normalized eigenfunctions are analytically obtained. The fine structure of the energy level for this system is also discussed.
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 $E=\pm M$.
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.
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.
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.
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.
14062967
Probing ~100 AU Intergalactic Mg II Absorbing “Cloudlets” with Quasar Microlensing
Intergalactic Mg II absorbers are known to have structures down to scales ~102.5 pc, and there are now indications that they may be fragmented on scales 10-2.5 pc according to Hao and coworkers. When a lensed quasar is microlensed, the microimages of the quasar experience creation, destruction, distortion, and drastic astrometric changes during caustic crossings. I show that quasar microlensing can effectively probe Mg II and other absorption "cloudlets" with sizes ~10-4.0 to 10-2.0 pc by inducing significant spectral variability on timescales of months to years. With numerical simulations, I demonstrate the feasibility of applying this method to Q2237+0305, and I show that high-resolution spectra of this quasar in the near future would provide a clear test of the existence of such metal-line absorbing "cloudlets."
118387861
Q in Other Solar Systems
A significant fraction of the hot Jupiters with final circularized orbital periods of less than 5 days are thought to form through the channel of high-eccentricity migration. Tidal dissipation at successive periastron passages removes orbital energy of the planet, which has the potential for changes in semi-major axis of a factor of ten to a thousand. In the equilibrium tide approximation we show that, in order for high-eccentricity migration to take place, the relative level of tidal dissipation in Jupiter analogues must be at least 10 times higher than the upper-limit attributed to the Jupiter-Io interaction. While this is not a severe problem for high-e migration, it contradicts the results of several previous calculations. We show that these calculations of high-e migration inadvertently over-estimated the strength of tidal dissipation by three to four orders of magnitude. These discrepancies were obscured by the use of various parameters, such as lag time \tau, tidal quality factor Q and viscous time t_V. We provide the values of these parameters required for the Jupiter-Io interaction, tidal circularization and high-e migration. Implications for tidal theory as well as models of the inflated radii of hot Jupiters are discussed. Though the tidal Q is not, in general, well-defined, we derive a formula for it during high-eccentricity migration where Q is approximately constant throughout evolution.
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.
18017754
The absolute magnitude of RRC variables from statistical parallax
We present the first definitive measurement of the absolute magnitude of RR Lyrae c-type variable stars (RRc) determined purely from statistical parallax. We use a sample of 242 RRc variables selected from the All Sky Automated Survey for which high-quality light curves, photometry, and proper motions are available. We obtain high-resolution echelle spectra for these objects to determine radial velocities and abundances as part of the Carnegie RR Lyrae Survey. We find that M V, RRc = 0.59 ? 0.10 at a mean metallicity of [Fe/H] = ?1.59. This is to be compared with previous estimates for RRab stars (M V, RRab = 0.76 ? 0.12) and the only direct measurement of an RRc absolute magnitude (RZ Cephei, M V, RRc = 0.27 ? 0.17). We find the bulk velocity of the halo relative to the Sun to be (W ?, W ?, Wz ) = (12.0, ?209.9, 3.0) km s?1 in the radial, rotational, and vertical directions with dispersions . For the disk, we find (W ?, W ?, Wz ) = (13.0, ?42.0, ?27.3) km s?1 relative to the Sun with dispersions . Finally, as a byproduct of our statistical framework, we are able to demonstrate that UCAC2 proper-motion errors are significantly overestimated as verified by UCAC4.
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.