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Andrea Sacchetti

Publications and source records attributed to Andrea Sacchetti.

At least 19 recordsLinked to original sources
arXiv2025

Mathematical results for the nonlinear Winter's model

In recent years, Winter's nonlinear model has been adopted in theoretical physics as the prototype for the study of quantum resonances and the dynamics of observables in the context of nonlinear Schr\"odinger equations. However, its mathematical treatment still has several important gaps. This article demonstrates a dispersive estimate of the evolution operator, from which the result of local well-posedeness of the solution follows; a criterion for the existence of the blow-up phenomenon is also provided. Finally, the phenomenon of bifurcations of stationary solutions is analysed, concluding with a conjecture on the orbital stability of some of them.

math-ph↗
arXiv2024

Resonances crossing effect and quantum sensor of electric fields

While the phenomenon of the exact crossing of energy levels is a rarely occurring event, in the case of quantum resonances associated with metastable states this phenomenon is much more frequent and various scenarios can occur. When there is an exact crossing of the imaginary parts of the resonances in a two-level quantum system subject to an external DC electric field, then a damped beating phenomenon occurs, which is absent if there is no such crossing. This fact, tested numerically on an explicit one-dimensional model, suggests the possibility of designing quantum sensors to determine in a very simple way whether the external field strength has an assigned value or not.

quant-ph↗
arXiv2023

Quantum resonances and analysis of the survival amplitude in the nonlinear Winter's model

In this paper we show that the typical effects of quantum resonances, namely, the exponential-type decay of the survival amplitude, continue to exist even when a nonlinear perturbative term is added to the time-dependent Schroedinger equation. The difficulty in giving a rigorous and appropriate definition of quantum resonances by means of the notions already used for linear equations is also highlighted.

math-ph↗
arXiv2022

Perturbation theory for nonlinear Schrodinger equations

Treating the nonlinear term of the Gross-Pitaevskii nonlinear Schrodinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh-Schrodinger power series. This power series is proved to be convergent when the parameter representing the intensity of the nonlinear term is less in absolute value than a threshold value, and it gives a stationary solution to the nonlinear Schrodinger equation.

math-ph↗
arXiv2022

Neural Networks to solve Partial Differential Equations: a Comparison with Finite Elements

We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM ($0.015$ for a solution on the order of unity) is easily achieved with NN too without much tuning of the hyperparameters. Accuracies below $0.01$ instead require refinement with an alternative optimizer to reach a similar performance with NN. A rough comparison between the Floating Point Operations values, as a machine-independent quantification of the computational performance, suggests a significant difference between FEM and NN in favour of the former. This also strongly holds for computation time: for an accuracy on the order of $10^{-5}$, FEM and NN require $54$ and $1100$ seconds, respectively. A detailed analysis of the effect of varying different hyperparameters shows that accuracy and computational time only weakly depend on the major part of them. Accuracies below $0.01$ cannot be achieved with the "adam'' optimizers and it looks as though accuracies below $10^{-5}$ cannot be achieved at all. In conclusion, the present work shows that for the concrete case of solving a steady-state 2D heat equation, the performance of a FEM algorithm is significantly better than the solution via networks.

cs.CE↗
arXiv2022

Spectral splitting method for nonlinear Schrödinger equations with quadratic potential

In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a one-dimensional nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.

math-ph↗
arXiv2021

Quantum forced oscillator via Wigner transform

In this paper we review the basic results concerning the Wigner transform and then we completely solve the quantum forced harmonic/inverted oscillator in such a framework; eventually, the tunnel effect for the forced inverted oscillator is discussed.

quant-ph↗
arXiv2020

Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations

Carlini's career was mainly dedicated to astronomy, but he was also a particularly skilled mathematician. In this article we collect and analyse his mathematical contributions in detail. In particular, in his important Memoir of the year 1817 devoted to Kepler's equation he introduced an innovative idea to solve ordinary differential equations with singular perturbations by means of asymptotic expansions. In the same Memoir also appeared, five years before Laplace's contributions, what is usually called the Laplace limit constant. Furthermore, Carlini published other mathematical Memoirs anticipating, 70 years in advance, the importance of complex branches of the Lambert's special function.

math.HO↗
arXiv2019

Derivation of the tight-binding approximation for time-dependent nonlinear Schrödinger equations

In this paper we consider the nonlinear one-dimensional time-dependent Schroedinger equation with a periodic potential and a local perturbation. In the limit of large periodic potential the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schroedinger equation of the tight-binding model.

math-ph↗
arXiv2018

Nonlinear Stark-Wannier equation

In this paper we consider stationary solutions to the nonlinear one-dimensional Schroedinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark-Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases.

math-ph↗
arXiv2017

Bifurcation trees of Stark-Wannier ladders for accelerated BECs in an optical lattice

In this paper we show that in the semicassical regime of periodic potential large enough, the Stark-Wannier ladders become a dense energy spectrum because of a cascade of bifurcations while increasing the ratio between the effective nonlinearity strength and the tilt of the external field; this fact is associated to a transition from regular to quantum chaotic dynamics. The sequence of bifurcation points is explicitly given.

quant-ph↗
arXiv2016

Bloch oscillations and accelerated Bose-Einstein condensates in an optical lattice

We discuss the method for the measurement of the gravity acceleration g by means of Bloch oscillations of an accelerated BEC in an optical lattice. This method has a theoretical critical point due to the fact that the period of the Bloch oscillations depends, in principle, on the initial shape of the BEC wavepacket. Here, by making use of the nearest-neighbor model for the numerical analysis of the BEC wavefunction, we show that in real experiments the period of the Bloch oscillations does not really depend on the shape of the initial wavepacket and that the relative uncertainty, due to the fact that the initial shape of the wavepacket may be asymmetrical, is smaller than the one due to experimental errors. Furthermore, we also show that the relation between the oscillation period and the scattering length of the BEC's atoms is linear; this fact suggest us a new experimental procedure for the measurement of the scattering length of atoms.

cond-mat.quant-gas↗
arXiv2015

Nonlinear Schrodinger equations with a multiple-well potential and a Stark-type perturbation

A Bose-Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross-Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrodinger equation. Then, by means of numerical experiments we show that the BEC's center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period.

math-ph↗
arXiv2015

Accelerated Bose-Einstein condensates in a double-well potential

Devices based on ultracold atoms moving in an accelerating optical lattice or double-well potential are a promising tool for precise measurements of fundamental physical constants as well as for the construction of sensors. Here, we carefully analyze the model of a couple of BECs separated by a barrier in an accelerated field and we show how the observable quantities, mainly the period of the beating motion or of the phase-shift, are related to the physical parameters of the model as well as to the energy of the initial state.

cond-mat.quant-gas↗
arXiv2014

Quantum resonances and time decay for a double-barrier model

Here we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of two repulsive Dirac's deltas. In such a "pedagogical" model we give, by means of the theory of quantum resonances, the explicit expression of the dominant terms of $<ψ, e^{-it H} ϕ>$, where $H$ is the double-barrier Hamiltonian operator and where $ψ$ and $ϕ$ are two test functions.

math-ph↗