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 $\Gamma(a,x)$ for any value of $a$ or $x$. 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to simple quantum--mechanical examples and show that under some conditions the main equations of the approach are no longer valid.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. (8

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 ($\approx 2000$).Comment: 12 pages, 12 figures, 2 table