2,309 research outputs found
Eigenvalue bounds for a class of singular potentials in N dimensions
The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963]
for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to
N-dimensions. In particular a simple formula is derived which bounds the
eigenvalues for the spiked harmonic oscillator potential V(x) = x^2 +
lambda/x^alpha, alpha > 0, lambda > 0, and is valid for all discrete
eigenvalues, arbitrary angular momentum ell, and spatial dimension N.Comment: 10 pages (plain tex with 2 ps figures). J.Phys.A:Math.Gen.(In Press
Variational analysis for a generalized spiked harmonic oscillator
A variational analysis is presented for the generalized spiked harmonic
oscillator Hamiltonian operator H, where H = -(d/dx)^2 + Bx^2+ A/x^2 +
lambda/x^alpha, and alpha and lambda are real positive parameters. The
formalism makes use of a basis provided by exact solutions of Schroedinger's
equation for the Gol'dman and Krivchenkov Hamiltonian (alpha = 2), and the
corresponding matrix elements that were previously found. For all the discrete
eigenvalues the method provides bounds which improve as the dimension of the
basis set is increased. Extension to the N-dimensional case in arbitrary
angular-momentum subspaces is also presented. By minimizing over the free
parameter A, we are able to reduce substantially the number of basis functions
needed for a given accuracy.Comment: 15 pages, 1 figur
Discrete Spectra of Semirelativistic Hamiltonians
We review various attempts to localize the discrete spectra of
semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r)
(w.l.o.g. in three spatial dimensions) as entering, for instance, in the
spinless Salpeter equation. Every Hamiltonian in this class of operators
consists of the relativistic kinetic energy \beta \sqrt{m^2 + p^2} (where \beta
> 0 allows for the possibility of more than one particles of mass m) and a
spherically symmetric attractive potential V(r), r = |x|. In general, accurate
eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of
a numerical approximation procedure. Our main emphasis, however, is on the
derivation of rigorous semi-analytical expressions for both upper and lower
bounds to the energy levels of such operators. We compare the bounds obtained
within different approaches and present relationships existing between the
bounds.Comment: 21 pages, 3 figure
Semiclassical energy formulas for power-law and log potentials in quantum mechanics
We study a single particle which obeys non-relativistic quantum mechanics in
R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2,
then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may
be represented exactly by the semiclassical expression E_{n\ell}(q) =
min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) =
ln(r). By writing one power as a smooth transformation of another, and using
envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are
monotone increasing. Recent refinements to the comparison theorem of QM in
which comparison potentials can cross over, allow us to prove for n = 1 that
Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q}
is monotone decreasing. Thus P(q) cannot increase too slowly. This result
yields some sharper estimates for power-potential eigenvlaues at the bottom of
each angular-momentum subspace.Comment: 20 pages, 5 figure
Gemini-South + FLAMINGOS Demonstration Science: Near-Infrared Spectroscopy of the z=5.77 Quasar SDSS J083643.85+005453.3
We report an infrared 1-1.8 micron (J+H-bands), low-resolution (R=450)
spectrogram of the highest-redshift radio-loud quasar currently known, SDSS
J083643.85+005453.3, obtained during the spectroscopic commissioning run of the
FLAMINGOS multi-object, near-infrared spectrograph at the 8m Gemini-South
Observatory. These data show broad emission from both CIV 1549 and CIII] 1909,
with strengths comparable to lower-redshift quasar composite spectra. The
implication is that there is substantial enrichment of the quasar environment,
even at times less than a billion years after the Big Bang. The redshift
derived from these features is z = 5.774 +/- 0.003, more accurate and slightly
lower than the z = 5.82 reported in the discovery paper based on the
partially-absorbed Lyman-alpha emission line. The infrared continuum is
significantly redder than lower-redshift quasar composites. Fitting the
spectrum from 1.0 to 1.7 microns with a power law f(nu) ~ nu^(-alpha), the
derived power law index is alpha = 1.55 compared to the average continuum
spectral index = 0.44 derived from the first SDSS composite quasar.
Assuming an SMC-like extinction curve, we infer a color excess of E(B-V) = 0.09
+/- 0.01 at the quasar redshift. Only approximately 6% of quasars in the
optically-selected Sloan Digital Sky Survey show comparable levels of dust
reddening.Comment: 10 pages, 1 figure; to appear in the Astrophysical Journal Letter
Perurbation expansions for the spiked harmonic oscillator and related series involving the gamma function
We study weak-coupling perturbation expansions for the ground-state energy of
the Hamiltonian with the generalized spiked harmonic oscillator potential V(x)
= Bx^2 + A/x^2 + lambda/x^alpha, and also for the bottoms of the angular
momentum subspaces labelled by ell = 0,1,2 ..., in N-dimensions corresponding
to the spiked harmonic oscillator potential: V(x) = x^2 + lambda/x^alpha, where
alpha is a real positive parameter. A method of Znojil is then applied to
obtain closed form expressions for the sums of some infinite series whose terms
involve ratios and products of gamma functions.Comment: 9 page
The BTC40 Survey for Quasars at 4.8 < z < 6
The BTC40 Survey for high-redshift quasars is a multicolor search using
images obtained with the Big Throughput Camera (BTC) on the CTIO 4-m telescope
in V, I, and z filters to search for quasars at redshifts of 4.8 < z < 6. The
survey covers 40 sq. deg. in B, V, & I and 36 sq. deg. in z. Limiting
magnitudes (3 sigma) reach to V = 24.6, I = 22.9 and z = 22.9. We used the
(V-I) vs. (I-z) two-color diagram to select high-redshift quasar candidates
from the objects classified as point sources in the imaging data. Follow-up
spectroscopy with the AAT and CTIO 4-m telescopes of candidates having I < 21.5
has yielded two quasars with redshifts of z = 4.6 and z = 4.8 as well as four
emission line galaxies with z = 0.6. Fainter candidates have been identified
down to I = 22 for future spectroscopy on 8-m class telescopes.Comment: 27 pages, 8 figures; Accepted for publication in the Astronomical
Journa
General energy bounds for systems of bosons with soft cores
We study a bound system of N identical bosons interacting by model pair
potentials of the form V(r) = A sgn(p)r^p + B/r^2, A > 0, B >= 0. By using a
variational trial function and the `equivalent 2-body method', we find explicit
upper and lower bound formulas for the N-particle ground-state energy in
arbitrary spatial dimensions d > 2 for the two cases p = 2 and p = -1. It is
demonstrated that the upper bound can be systematically improved with the aid
of a special large-N limit in collective field theory
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