1,748 research outputs found
Asymptotic and exact series representations for the incomplete Gamma function
Using a variational approach, two new series representations for the
incomplete Gamma function are derived: the first is an asymptotic series, which
contains and improves over the standard asymptotic expansion; the second is a
uniformly convergent series, completely analytical, which can be used to obtain
arbitrarily accurate estimates of for any value of or .
Applications of these formulas are discussed.Comment: 8 pages, 4 figure
The period of a classical oscillator
We develop a simple method to obtain approximate analytical expressions for
the period of a particle moving in a given potential. The method is inspired to
the Linear Delta Expansion (LDE) and it is applied to a large class of
potentials. Precise formulas for the period are obtained.Comment: 5 pages, 4 figure
Further analysis of the connected moments expansion
We apply the connected moments expansion to simple quantum--mechanical
examples and show that under some conditions the main equations of the approach
are no longer valid. In particular we consider two--level systems, the harmonic
oscillator and the pure quartic oscillator.Comment: 19 pages; 2 tables; 4 figure
Comment on: "Analytical approximations for the collapse of an empty spherical bubble"
We analyze the Rayleigh equation for the collapse of an empty bubble and
provide an explanation for some recent analytical approximations to the model.
We derive the form of the singularity at the second boundary point and discuss
the convergence of the approximants. We also give a rigorous proof of the
asymptotic behavior of the coefficients of the power series that are the basis
for the approximate expressions
Bound states for the quantum dipole moment in two dimensions
We calculate accurate eigenvalues and eigenfunctions of the Schr\"odinger
equation for a two-dimensional quantum dipole. This model proved useful for the
study of elastic effects of a single edge dislocation. We show that the
Rayleigh-Ritz variational method with a basis set of Slater-type functions is
considerably more efficient than the same approach with the basis set of
point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in
earlier calculations
Small-energy series for one-dimensional quantum-mechanical models with non-symmetric potentials
We generalize a recently proposed small-energy expansion for one-dimensional
quantum-mechanical models. The original approach was devised to treat symmetric
potentials and here we show how to extend it to non-symmetric ones. Present
approach is based on matching the logarithmic derivatives for the left and
right solutions to the Schr\"odinger equation at the origin (or any other point
chosen conveniently) . As in the original method, each logarithmic derivative
can be expanded in a small-energy series by straightforward perturbation
theory. We test the new approach on four simple models, one of which is not
exactly solvable. The perturbation expansion converges in all the illustrative
examples so that one obtains the ground-state energy with an accuracy
determined by the number of available perturbation corrections
Comment on: `Numerical estimates of the spectrum for anharmonic PT symmetric potentials' [Phys. Scr. \textbf{85} (2012) 065005]
We show that the authors of the commented paper draw their conclusions from
the eigenvalues of truncated Hamiltonian matrices that do not converge as the
matrix dimension increases. In one of the studied examples the authors missed
the real positive eigenvalues that already converge towards the exact
eigenvalues of the non-Hermitian operator and focused their attention on the
complex ones that do not. We also show that the authors misread Bender's
argument about the eigenvalues of the harmonic oscillator with boundary
conditions in the complex- plane (Rep. Prog. Phys. {\bf 70} (2007) 947).Comment: 7 pages, 1 tabl
Particle correlation from uncorrelated non Born-Oppenheimer SCF wavefunctions
We analyse a nonadiabatic self-consistent field method by means of an
exactly-solvable model. The method is based on nuclear and electronic orbitals
that are functions of the cartesian coordinates in the laboratory-fixed frame.
The kinetic energy of the center of mass is subtracted from the molecular
Hamiltonian operator in the variational process. The results for the simple
model are remarkably accurate and show that the integration over the redundant
cartesian coordinates leads to couplings among the internal ones
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