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Paolo Amore

Publications and source records attributed to Paolo Amore.

At least 91 records · Page 5Linked to original sources
arXiv2011

Comment on "Bound states of edge dislocations: The quantum dipole problem in two dimensions"

We show that the numerical results contained in a recent paper are affected by a non optimal implementation of the methods which have been used to obtain these results. A careful analysis done using the Rayleigh-Ritz method provides a rigorous upper bound for the energy of the ground state of an electron in a two dimensional potential generated by the edge dislocation, as well as precise values for the excited states. The extrapolation of the results corresponding to different subspaces is used to obtain a precise estimate of the fundamental energy of the model. The energies of the first 500 states that we have calculated are in perfect agreement with the expected asymptotic behavior.

cond-mat.mtrl-sci↗
arXiv2010

High--order connected moments expansion for the Rabi Hamiltonian

We analyze the convergence properties of the connected moments expansion (CMX) for the Rabi Hamiltonian. To this end we calculate the moments and connected moments of the Hamiltonian operator to a sufficiently large order. Our large--order results suggest that the CMX is not reliable for most practical purposes because the expansion exhibits considerable oscillations.

quant-ph↗
arXiv2010

The string of variable density: further results

We analyze the problem of calculating the solutions and the spectrum of a string with arbitrary density and fixed ends. We build a perturbative scheme which uses a basis of WKB-type functions and obtain explicit expressions for the eigenvalues and eigenfunctions of the string. Using this approach we show that it is possible to derive the asymptotic (high energy) behavior of the string, obtaining explicit expressions for the first three coefficients (the first two can also be obtained with the WKB method). Finally using an iterative approach we also obtain analytical expressions for the low energy behavior of the eigenvalues and eigenfunctions of a string with rapidly oscillating density, recovering (in a simpler way) results in the literature.

math-ph↗
arXiv2010

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.

math-ph↗
arXiv2010

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 ($\approx 2000$).

quant-ph↗
arXiv2009

Collocation method for fractional quantum mechanics

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on Little Sinc Functions (LSF), which discretizes the Schrödinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a WKB analysis is performed.

quant-ph↗
arXiv2009

Can one hear the density of a drum? Weyl's law for inhomogeneous media

We generalize Weyl's law to inhomogeneous bodies in $d$ dimensions. Using a perturbation scheme recently obtained by us in Ref. \cite{Amore09}, we have derived an explicit formula, which describes the asymptotic behavior of the eigenvalues of the negative laplacian on a closed $d$-dimensional cubic domain, either with Dirichlet or Neumann boundary conditions. For homogeneous bodies, the leading term in our formula reduces to the standard expression for Weyl's law. We have also used Weyl's conjecture to obtain a non-perturbative extension of our formula and we have compared our analytical results with the precise numerical results obtained using the Conformal Collocation Method of Refs. \cite{Amore08,Amore09}.

math-ph↗
arXiv2009

Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results

We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.

quant-ph↗
arXiv2009

Harmonic oscillator in a one-dimensional box

We study a harmonic molecule confined to a one--dimensional box with impenetrable walls. We explicitly consider the symmetry of the problem for the cases of different and equal masses. We propose suitable variational functions and compare the approximate energies given by the variation method and perturbation theory with accurate numerical ones for a wide range of values of the box length. We analyze the limits of small and large box size.

quant-ph↗
arXiv2009

Variational collocation for systems of coupled anharmonic oscillators

We have applied a collocation approach to obtain the numerical solution to the stationary Schrödinger 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.

quant-ph↗
arXiv2009

The virial theorem for nonlinear problems

We show that the virial theorem provides a useful simple tool for approximating nonlinear problems. In particular we consider conservative nonlinear oscillators and a bifurcation problem. In the former case we obtain the same main result derived earlier from the expansion in Chebyshev polynomials.

math-ph↗
arXiv2009

He's amazing calculations with the Ritz method

We discuss an earlier application of the Ritz variational method for strongly nonlinear problems. We clearly prove that the results derived for several extremely simple problems of supposedly physical and mathematical interest do not provide any clue on the utility of the approach.

math-ph↗
arXiv2008

The Schrödinger equation with piercings

We show that the spectrum of the Schrödinger equation in two or higher dimensions does not change when Dirichlet boundary conditions are enforced on a number of isolated points inside the original domain (piercings). We have obtained the analytical solution for spherically symmetric state of the $d$-dimensional simple harmonic oscillator pierced at the origin. Results for the case with multiple piercings are obtained numerically and agree with the theoretical prediction. In the case of a two dimensional parabolic quantum dot with two electrons and a single piercing in the origin we show that the energy of the ground state calculated to first order in perturbation theory goes over to the equivalent result in absence of piercing as the radius of the piercing becomes infinitesimal. Interestingly, we find that the leading finite size correction to the interaction energy is negative while the corresponding correction to the single particle energies is (as expected) positive. For a finite radius of the piercing and above a critical coupling, the first term dominates the second and the total energy of the dot with piercing is lower. Unfortunately the critical coupling found for this example is nonperturbative. Finally we study configurations allowing bound states in the continuum, showing that bound states survive to the insertion of piercings and that their energy is unchanged when the radius of the piercing vanishes.

cond-mat.other↗
arXiv2008

Discrete Higgs and the Cosmological Constant

It is proposed that the Higgs vacuum possesses a small-scale structure that can explain the large discrepancy between the predicted electroweak vacuum energy density and the observed cosmological constant. An effective Lagrangian description is employed to obtain modifications to the Standard Model predictions that can be tested at collider experiments.

hep-ph↗
arXiv2008

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.

physics.comp-ph↗
arXiv2008

Solving the Helmholtz equation for membranes of arbitrary shape

I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples,part of them taken from the literature, and part of them new. When possible, I have testedthe accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the L-shaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.

physics.comp-ph↗
arXiv2007

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.

math-ph↗