1,075 research outputs found
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
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
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are no longer valid.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
Inversion of perturbation series
We investigate the inversion of perturbation series and its resummation, and
prove that it is related to a recently developed parametric perturbation
theory. Results for some illustrative examples show that in some cases series
reversion may improve the accuracy of the results
Weakly bound states in heterogeneous waveguides
We study the spectrum of the Helmholtz equation in a two-dimensional infinite
waveguide, containing a weak heterogeneity localized at an internal point, and
obeying Dirichlet boundary conditions at its border. We prove that, when the
heterogeneity corresponds to a locally denser material, the lowest eigenvalue
of the spectrum falls below the continuum threshold and a bound state appears,
localized at the heterogeneity. We devise a rigorous perturbation scheme and
derive the exact expression for the energy to third order in the heterogeneity.Comment: 14 page
Variational collocation for systems of coupled anharmonic oscillators
We have applied a collocation approach to obtain the numerical solution to
the stationary Schr\"odinger equation for systems of coupled oscillators. The
dependence of the discretized Hamiltonian on scale and angle parameters is
exploited to obtain optimal convergence to the exact results. A careful
comparison with results taken from the literature is performed, showing the
advantages of the present approach.Comment: 14 pages, 10 table
Accurate calculation of the solutions to the Thomas-Fermi equations
We obtain highly accurate solutions to the Thomas-Fermi equations for atoms
and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method,
numerical integration, power series with Pad\'e and Hermite-Pad\'e approximants
and Chebyshev polynomials. Both the slope at origin and the location of the
right boundary in the magnetic-field case are given with unprecedented
accuracy
Non perturbative regularization of one loop integrals at finite temperature
A method devised by the author is used to calculate analytical expressions
for one loop integrals at finite temperature. A non-perturbative regularization
of the integrals is performed, yielding expressions of non-polynomial nature. A
comparison with previuosly published results is presented and the advantages of
the present technique are discussed.Comment: 7 pages, 2 figures, 2 tables; corrected some typos and simplified eq.
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Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results
We obtain systematic approximations to the states (energies and wave
functions) of quantum rings (annular drums) of arbitrary shape by conformally
mapping the annular domain to a simply connected domain. Extending the general
results of Ref.\cite{Amore09} we obtain an analytical formula for the spectrum
of quantum ring of arbirtrary shape: for the cases of a circular annulus and of
a Robnik ring considered here this formula is remarkably simple and precise. We
also obtain precise variational bounds for the ground state of different
quantum rings. Finally we extend the Conformal Collocation Method of
\cite{Amore08,Amore09} to the class of problems considered here and calculate
precise numerical solutions for a large number of states ().Comment: 12 pages, 12 figures, 2 table
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