121535344
EXACT QUANTIZATION RULE AND ITS APPLICATIONS TO PHYSICAL POTENTIALS
By using the exact quantization rule, we present analytical solutions of the Schrodinger equation for the deformed harmonic oscillator in one dimension, the Kratzer potential and pseudoharmonic oscillator in three dimensions. The energy levels of all the bound states are easily calculated from this quantization rule. The normalized wavefunctions are also obtained. It is found that the present approach can simplify the calculations.
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.
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.
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 $E=\pm M$.
118383144
SUPER-ECCENTRIC MIGRATING JUPITERS
An important class of formation theories for hot Jupiters involves the excitation of extreme orbital eccentricity (e = 0.99 or even larger) followed by tidal dissipation at periastron passage that eventually circularizes the planetary orbit at a period less than 10 days. In a steady state, this mechanism requires the existence of a significant population of super-eccentric (e > 0.9) migrating Jupiters with long orbital periods and periastron distances of only a few stellar radii. For these super-eccentric planets, the periastron is fixed due to conservation of orbital angular momentum and the energy dissipated per orbit is constant, implying that the rate of change in semi-major axis a is a-dot {proportional_to}a{sup 1/2} and consequently the number distribution satisfies dN/d log a{proportional_to}a{sup 1/2}. If this formation process produces most hot Jupiters, Kepler should detect several super-eccentric migrating progenitors of hot Jupiters, allowing for a test of high-eccentricity migration scenarios.
54818607
GRB 091029: at the limit of the fireball scenario
Aims. Using high-quality, broad-band afterglow data for GRB 091029, we test the validity of the forward-shock model for gamma-ray burst afterglows. Methods. We used multi-wavelength (NIR to X-ray) follow-up observations obtained with the GROND, BOOTES-3/YA and Stardome optical ground-based telescopes, and the UVOT and the XRT onboard the Swift satellite. The resulting data of excellent accuracy allow us to construct a multi-wavelength light curve with relative photometric errors as low as 1%, as well as the well-sampled spectral energy distribution covering 5 decades in energy. Results. The optical/NIR and the X-ray light curves of the afterglow of GRB 091029 are almost totally decoupled. The X-ray light curve shows a shallow rise with a peak at ∼7 ks and a decay slope of α ∼ 1.2 afterwards, while the optical/NIR light curve shows a much steeper early rise with a peak around 400 s, followed by a shallow decay with temporal index of α ∼ 0.6, a bump and a steepening of the decay afterwards. The optical/NIR spectral index decreases gradually by over 0.3 before this bump, and then slowly increases again, while the X-ray spectral index remains constant throughout the observations. Conclusions. To explain the decoupled light curves in the X-ray and optical/NIR domains, a two-component outflow is proposed. Several models are tested, including continuous energy injection, components with different electron energy indices and components in two different stages of spectral evolution. Only the last model can explain both the decoupled light curves with asynchronous peaks and the peculiar SED evolution. However, this model has so many unknown free parameters that we are unable to reliably confirm or disprove its validity, making the afterglow of GRB 091029 difficult to explain in the framework of the simplest fireball model. This conclusion provides evidence that a scenario beyond the simplistic assumptions is needed to be able to model the growing number of well-sampled afterglow light curves.
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.
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.
119183924
Chemical evolution of the Galactic bulge as traced by microlensed dwarf and subgiant stars - III. Detection of lithium in the metal-poor bulge dwarf MOA-2010-BLG-285S
Context. To study the evolution of Li in the Galaxy it is necessary to observe dwarf or subgiant stars. These are the only long-lived stars whose present-day atmospheric chemical composition reflects their natal Li abundances according to standard models of stellar evolution. Although Li has been extensively studied in the Galactic disk and halo, to date there has only been one uncertain detection of Li in an unevolved bulge star. Aims. Our aim with this study is to provide the first clear detection of Li in the Galactic bulge, based on an analysis of a dwarf star that has largely retained its initial Li abundance. Methods. We performed a detailed elemental abundance analysis of the bulge dwarf star MOA-2010-BLG-285S using a high-resolution and high signal-to-noise spectrum obtained with the UVES spectrograph at the VLT when the object was optically magnified during a gravitational microlensing event (visual magnification A similar to 550 during observation). The Li abundance was determined through synthetic line profile fitting of the Li-7 resonance doublet line at 670.8 nm. The results have been corrected for departures from LTE. Results. MOA-2010-BLG-285S is, at [Fe/H] = -1.23, the most metal-poor dwarf star detected so far in the Galactic bulge. Its old age (12.5 Gyr) and enhanced [alpha/Fe] ratios agree well with stars in the thick disk at similar metallicities. This star represents the first unambiguous detection of Li in a metal-poor dwarf star in the Galactic bulge. We find an NLTE corrected Li abundance of log epsilon(Li) = 2.16, which is consistent with values derived for Galactic disk and halo dwarf stars at similar metallicities and temperatures. Conclusions. Our results show that there are no signs of Li enrichment or production in the Galactic bulge during its earliest phases. Observations of Li in other galaxies (omega Cen) and other components of the Galaxy suggest further that the Spite plateau is universal. (Less)