118042719
Levinson's Theorem for the Schrödinger Equation in One Dimension
Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx−2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn+(n−) is related to the phase shift η+ (0)[η− (0)] of the scattering states with the same parity at zero momentum as η+ (0)+ π/2 =n+ π and η− (0) =n− πfor the noncritical case, and η+ (0) =n+ π andη− (0) − π/2 =n− π for the critical case.
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.
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.
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.
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.
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
20954769
MOA-2009-BLG-387Lb: A massive planet orbiting an M dwarf
Aims. We report the discovery of a planet with a high planet-to-star mass ratio in the microlensing event MOA-2009-BLG-387, which exhibited pronounced deviations over a 12-day interval, one of the longest for any planetary event. The host is an M dwarf, with a mass in the range 0.07 M_⊙ < M_(host) < 0.49 M_⊙ at 90% confidence. The planet-star mass ratio q = 0.0132 ± 0.003 has been measured extremely well, so at the best-estimated host mass, the planet mass is m_p = 2.6 Jupiter masses for the median host mass, M = 0.19 M_⊙. Methods. The host mass is determined from two “higher order” microlensing parameters. One of these, the angular Einstein radius θ_E = 0.31 ± 0.03 mas has been accurately measured, but the other (the microlens parallax πE, which is due to the Earth’s orbital motion) is highly degenerate with the orbital motion of the planet. We statistically resolve the degeneracy between Earth and planet orbital effects by imposing priors from a Galactic model that specifies the positions and velocities of lenses and sources and a Kepler model of orbits. Results. The 90% confidence intervals for the distance, semi-major axis, and period of the planet are 3.5 kpc < D_L < 7.9 kpc, 1.1 AU < a < 2.7 AU, and 3.8 yr < P < 7.6 yr, respectively.
118452553
The Effect of Differential Limb Magnification on Abundance Analysis of Microlensed Dwarf Stars
Finite source effects can be important in observations of gravitational microlensing of stars. Near caustic crossings, for example, some parts of the source star will be more highly magnified than other parts. The spectrum of the star is then no longer the same as when it is unmagnified, and measurements of the atmospheric parameters and abundances will be affected. The accuracy of abundances measured from spectra taken during microlensing events has become important recently because of the use of highly magnified dwarf stars to probe abundance ratios and the abundance distribution in the Galactic bulge. In this paper, we investigate the importance of finite source effects on spectra by using magnification profiles motivated by two events to synthesize spectra for dwarfs between 5000 K and 6200 K at solar metallicity. We adopt the usual techniques for analyzing the microlensed dwarfs, namely, spectroscopic determination of temperature, gravity, and microturbulent velocity, relying on equivalent widths. We find that ignoring the finite source effects for the more extreme case results in errors in T eff <45 K, in log g of <0.1 dex, and in ξ of <0.1 km s–1. In total, changes in equivalent widths lead to small changes in atmospheric parameters and changes in abundances of <0.06 dex, with changes in [Fe I/H] of <0.03 dex. For the case with a larger source-lens separation, the error in [Fe I/H] is <0.01 dex. This latter case represents the maximum effect seen in events whose light curves are consistent with a point-source lens, which includes the majority of microlensed bulge dwarfs published so far.