A new method for studying the vibration of non-homogeneous membranes

We present a method to solve the Helmholtz equation for a non-homogeneous membrane with Dirichlet boundary conditions at the border of arbitrary two-dimensional domains. The method uses a collocation approach based on a set of localized functions, called "little sinc functions", which are used to discretize two-dimensional regions. We have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature. Our results show that the present method is very accurate and that its implementation for general problems is straightforward.Comment: 16 pages, 7 figures, 6 table

Neutron and muon-induced background studies for the AMoRE double-beta decay experiment

AMoRE (Advanced Mo-based Rare process Experiment) is an experiment to search a neutrinoless double-beta decay of $^{100}$Mo in molybdate crystals. The neutron and muon-induced backgrounds are crucial to obtain the zero-background level (<$10^{-5}$ counts/(keV$\cdot$kg$\cdot$yr)) for the AMoRE-II experiment, which is the second phase of the AMoRE project, planned to run at YEMI underground laboratory. To evaluate the effects of neutron and muon-induced backgrounds, we performed Geant4 Monte Carlo simulations and studied a shielding strategy for the AMORE-II experiment. Neutron-induced backgrounds were also included in the study. In this paper, we estimated the background level in the presence of possible shielding structures, which meet the background requirement for the AMoRE-II experiment

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

The string of variable density: perturbative and non-perturbative results

We obtain systematic approximations for the modes of vibration of a string of variable density, which is held fixed at its ends. These approximations are obtained iteratively applying three theorems which are proved in the paper and which hold regardless of the inhomogeneity of the string. Working on specific examples we obtain very accurate approximations which are compared both with the results of WKB method and with the numerical results obtained with a collocation approach. Finally, we show that the asymptotic behaviour of the energies of the string obtained with perturbation theory, worked to second order in the inhomogeinities, agrees with that obtained with the WKB method and implies a different functional dependence on the density that in two and higher dimensions.Comment: 28 pages, 3 tables, 6 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