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Andrea SACCHETTI

Professore Ordinario
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2023 - Enhancement of the Zakharov–Glassey’s method for Blow-up in nonlinear Schrödinger equations [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we give a sharper sufficient condition for blow-up of the solution to a nonlinear Schrodinger equation with free/Stark/quadratic potential by improving the well known Zakharov–Glassey's method.

2023 - Perturbation theory for nonlinear Schrödinger equations [Articolo su rivista]
Sacchetti, Andrea
abstract

Treating the nonlinear term of the Gross–Pitaevskii nonlinear Schrödinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh–Schrödinger 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 Schrödinger equation.

2023 - Quantum forced oscillator via Wigner transform [Articolo su rivista]
Sacchetti, A.
abstract

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

2023 - Tunnel effect and analysis of the survival amplitude in the nonlinear Winter’s model [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we show how the nonlinearity term affects the tunnel effect and the survival amplitude in the nonlinear Win- ter’s model. In particular, in the case of attractive nonlinearity large enough it turns out that the tunnel effect is going to disappear. Furthermore, 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.

2022 - Spectral splitting method for nonlinear Schrödinger equations with quadratic potential [Articolo su rivista]
Sacchetti, Andrea
abstract

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 Schrödinger 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.

2020 - Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schrödinger Equations [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper, we consider the nonlinear one-dimensional timedependent Schr¨odinger equation with a periodic potential and a bounded 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.

2020 - Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations [Articolo su rivista]
Sacchetti, Andrea
abstract

2020 - On a mathematical model for a damped and driven double-well Bose–Einstein condensate [Articolo su rivista]
Gavioli, A.; Sacchetti, A.
abstract

In this paper we consider a two-mode dynamical system as a model for a driven one-dimensional damped Bose–Einstein condensate in a double-well trapping potential. In the case of a constant external driving force the existence and stability of stationary solutions are discussed in relation to the values of the physical parameters. In the case of a time-dependent periodic external driving force the existence of limit cycles is proved, and the amplitude of these limit cycles exhibits a jump phenomenon for critical values of the physical parameters.

2020 - Stationary solutions to cubic nonlinear Schrödinger equations with quasi-periodic boundary conditions [Articolo su rivista]
Sacchetti, A.
abstract

In this paper we give the quantization rules to determine the normalized stationary solutions to the cubic nonlinear Schrödinger equation with quasi-periodic conditions on a given interval. Similarly to what happen in the Floquet's theory for linear periodic operators, also in this case some kind of band functions there exist.

2019 - Nonlinear models and bifurcation trees in quantum mechanics: a review of recent results [Articolo su rivista]
Sacchetti, Andrea
abstract

In this talk we discuss some recent results I obtained for a class of nonlinear models in quantum mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger 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; for a detailed treatment we refer to Sacchetti (Phys Rev E 95:062212, 2017, SIAM J Math Anal 50(6):5783–5810, 2018) where this model has been studied.

2018 - Nonlinear Stark--Wannier Equation [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we consider stationary solutions to the nonlinear one-dimensional Schrödinger 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.

2018 - On the introduction of singular potentials in Quantum Mechanics - A contribution by Enrico Fermi | [Sull’introduzione dei potenziali singolari in Meccanica Quantistica - un contributo di Enrico Fermi] [Articolo su rivista]
Sacchetti, Andrea
abstract

La teoria degli operatori lineari con potenziali singolari ha ormai acquisito all'interno dei Metodi Matematici per la Meccanica Quantistica un ruolo ben definito e consolidato. In tale teoria gioca un ruolo importante la distribuzione delta di Dirac. Con questo articolo si cerca di individuare la genesi di questo filone di ricerca e, in modo più specifico, riconoscere l'importante contributo dato dallo scienziato italiano Enrico Fermi.

2017 - Bifurcation trees of Stark-Wannier ladders for accelerated Bose-Einstein condensates in an optical lattice [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we show that in the semiclassical 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.

2017 - Bloch oscillations and accelerated Bose–Einstein condensates in an optical lattice [Articolo su rivista]
Sacchetti, Andrea
abstract

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 suggests us a new experimental procedure for the measurement of the scattering length of atoms.

2017 - Double-barrier resonances and time decay of the survival probability: A toy model [Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract

In this talk we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of repulsive Dirac’s deltas. In such a pedagogical model we give, by means of the theory of quantum resonances, the asymptotic behavior of (Formula Found) for large times, where H is the double-barrier Hamiltonian operator and where ψ and ϕ are two test functions. In particular, when ψ is close to a resonant state then explicit expression of the dominant terms of the survival probability defined as (Formula Found) is given.

2016 - Accelerated Bose-Einstein condensates in a double-well potential [Articolo su rivista]
Sacchetti, Andrea
abstract

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.

2016 - Nonlinear Schrödinger equations with a multiple-well potential and a Stark-type perturbation [Articolo su rivista]
Sacchetti, Andrea
abstract

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 Schrödinger 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.

2016 - Quantum resonances and time decay for a double-barrier model [Articolo su rivista]
Sacchetti, Andrea
abstract

Here we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of two repulsive Diracs 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-itH φ〉, where H is the double-barrier Hamiltonian operator and where ψ and f are two test functions.

2015 - Solution to the double-well nonlinear Schrödinger equation with Stark-type external field [Articolo su rivista]
Sacchetti, Andrea
abstract

Here we consider one- and two-dimensional nonlinear Schrödinger equations with double well potential and a Stark-type perturbation term. In the semi-classical limit we give an explicit solution to these equations for times of the order of the unperturbed beating period, up to an exponentially small remainder term. In particular, it turns out that the solution has a periodic behavior and the period is explicitly computed.

2014 - Existence of the Stark-Wannier quantum resonances [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper, we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrödinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.

2014 - First principle explanation of phase transition for Bose-Einstein condensates in optical lattices [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we consider Bose-Einstein condensates (BECs) in one-, two- and three-dimension lattice potentials. The key argument for the explanation of the transition from Superfluidity phase to Mott- Insulator phase is suggested to be the spontaneous symmetry breaking effect which occurs for critical values of the ratio between the on-site interaction term and the hopping matrix element. Such an effect can be directly seen in the Gross-Pitaevskii equation with double-well potentials and it also explains the different behavior between one-dimensional models and two/three-dimensional models.

2014 - Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials [Articolo su rivista]
R., Fukuizumi; Sacchetti, Andrea
abstract

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed

2014 - Stationary solutions to the multi-dimensional Gross–Pitaevskii equation with double-well potential [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we consider a nonlinear Schr ̈odinger equation with a cubic nonlinearity and a multi-dimensional double well potential. In the semiclassical limit the problem of the existence of stationary solutions simply reduces to the analysis of a finite dimensional Hamiltonian system which exhibits different behaviour depending on the dimension. In particular, in dimension 1 the symmetric stationary solution shows a standard pitchfork bifurcation effect, while in dimensions 2 and 3 new asymmetrical solutions associated with saddle points occur. These last solutions are localized on a single well and this fact is related to the phase transition effect observed in Bose–Einstein condensates in periodical lattices.

2012 - Nonlinear Schrödinger equations with multiple-well potential [Articolo su rivista]
Sacchetti, Andrea
abstract

We consider the stationary solutions for a class of Schrödinger equations with a N-well potential and a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of the ground state solutions is described by a N-dimensional Hamiltonian system, where the coupling term among the coordinates is a tridiagonal Toeplitz matrix. In particular, in the limit of large focusing nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on a single well. Furthermore, we consider in detail the case of N = 4 wells, where we show the occurrence of spontaneous symmetry-breaking bifurcation effect.

2011 - Bifurcation and Stability for Nonlinear SchrödingerEquations with DoubleWell Potential in the SemiclassicalLimit [Articolo su rivista]
R., Fukuizumi; Sacchetti, Andrea
abstract

We consider the stationary solutions for a class of Schrödinger equations witha symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassicallimit we prove that the reduction to a finite-mode approximation give the stationary solutions,up to an exponentially small term, and that symmetry-breaking bifurcation occurs ata given value for the strength of the nonlinear term. The kind of bifurcation picture onlydepends on the nonlinearity power. We then discuss the stability/instability properties ofeach branch of the stationary solutions. Finally, we consider an explicit one-dimensional toymodel where the double well potential is given by means of a couple of attractive Dirac’sdelta pointwise interactions.

2011 - Effect of quasibound states on coherent electron transport in twisted nanowires [Articolo su rivista]
G., Cuoghi; Bertoni, Andrea; Sacchetti, Andrea
abstract

Quantum transmission spectra of a twisted electron waveguide expose the coupling between traveling andquasibound states. Through a direct numerical solution of the open-boundary Schroedinger equation, we singleout the effects of the twist and show how the presence of a localized state leads to a Breit-Wigner or a Fanoresonance in the transmission.We also find that the energy of quasibound states is increased by the twist, despitethe constant section area along the waveguide. While the mixing of different transmission channels is expectedto reduce the conductance, the shift of localized levels into the traveling-states energy range can reduce theirdetrimental effects on coherent transport.

2011 - Resonant states for a three-body problem under an external field [Articolo su rivista]
V., Grecchi; H., Kovarik; A., Martinez; Sacchetti, Andrea; A., Sordoni
abstract

Here we consider one of the basic models for many-body problems under an external field: the molecule ion H2 under the effect of an external Stark-type potential. If we consider the vibrational energy levels of the first two electronic states of the molecule ion H2 then, in the semiclassical limit and by means of a suitable modified Born–Oppenheimer method, we can prove that they switch to sharp resonances localized in the same interval of energy of the vibrational levels when an external Stark-type field, with the same direction of the nuclear axis, occurs.

2010 - A nonlinear Schrodinger equation with two symmetric point interactions in one dimension [Articolo su rivista]
Hynek, Kovarık; Sacchetti, Andrea
abstract

We consider a time-dependent one-dimensional nonlinear Schrodinger equation with a symmetric double-well potential represented by two Dirac’s δ. Among our results we give an explicit formula for the integral kernel of the unitarysemigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem.

2010 - Electrical current in nanoelectronic devices [Articolo su rivista]
Sacchetti, Andrea
abstract

In ultra-small electronic devices of the next generations the semiclassical model of electron motion in a periodical lattice between collisions turns out to be inadequate because the electron spread has magnitude order of the size of the ultra-small electronic device. In this Letter we consider the basic conceptual framework regarding how the length scale of the electrical device influences the transport behavior of the electrons between collisions and the electrical current. By taking into account the interference effects we obtain a very basic model for electrons transport, where the density current peak is given as function on the ratio between the thermal de Broglie wavelength and the lattice period. This result could be also useful in order to understand the basic effect of the insulator/metal transition.

2010 - Hysteresis effects in Bose-Einstein condensates [Articolo su rivista]
Sacchetti, Andrea
abstract

Here, we consider damped two-component Bose-Einstein condensates with many-body interactions. We show that, when the external trapping potential has a double-well shape and when the nonlinear coupling factors are modulated in time, hysteresis effects may appear under some circumstances. Such hysteresis phenomena are a result of the joint contribution of the appearance of saddle node bifurcations and the damping effect.

2010 - On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects [Capitolo/Saggio]
R., Adami; D., Noja; Sacchetti, Andrea
abstract

Bose-Einstein condensation and the related topic of Gross-Pitaevskii equation have become an important source of models and problems in mathematical physics and analysis. In particular, in the last decade, the interest in low-dimensional systems that evolve through the nonlinear Schroedinger equation has undergone an impressive growth. The reason is twofold: on the one hand, effectively one-dimensional Bose-Einstein condensates are currently realized, and the investigation on their dynamics isnowadays a well-developed field for experimentalists. On the other hand, in contrast to its higher-dimensional analogous,the one-dimensional nonlinear Schroedinger equation allows explicit solutions, that simplify remarkably the analysis. The recentliterature reveals an increasing interest for the dynamics ofnonlinear systems in the presence of so-called defects, namelymicroscopic scatterers, which model the presence of impurities.We review here some recent achievements on such systems, withparticular attention to the cases of the ``Dirac's delta'' and ``delta prime'' defects. We give rigorous definitions, recall and comment on known results for the delta case, and introduce new results for the delta prime case. The latter system turns out to be richer and interesting since it produces a bifurcation with symmetry breaking in the ground state.Our purpose lies mainly on collecting and conveying results, so proofs are not included.

2009 - Perturbation Theory, Semiclassical [Voce in Dizionario o Enciclopedia]
Sacchetti, Andrea
abstract

2009 - Third School and Workshop on "Mathematical Methods in Quantum Mechanics" [Esposizione]
Sacchetti, Andrea
abstract

The aim of the meeting is to present the state of the art in some challenging open problems in Quantum Mechanics from the point of view of Mathematical Physics. It is mainly addressed to young people interested in working on the subject.Among the topics covered: quantum systems with magnetic fields, quantum transport theory, quantum dechoerence and entanglement, classical behaviour in quantum systems, scattering and spectral analysis for Schroedinger operators, quantum chaos, adiabatic and semiclassical methods. non linear Schroedinger equations.Three courses will be given in a series of lectures scheduled in the morning of each day. Some invited talks will be given in the afternoon followed by short contributed talks given by participants.

2009 - Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential [Articolo su rivista]
Sacchetti, Andrea
abstract

Here we consider stationary states for nonlinear Schrödinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.

2008 - Bloch oscillators in a slowly perturbed external field [Articolo su rivista]
Sacchetti, Andrea
abstract

A quantum particle in a periodical lattice under the effect of an external homogeneous field shows a periodical motion, usually a named Bloch oscillator, for long times. When we introduce a weak and slowly varying inhomogeneous field then the dynamics of the quantum particle still exhibits a periodical motion but with a different period and a different width of the interval of oscillation. In this paper we obtain a formula for the dominant terms of the perturbed period and width, then we apply our result to the study of the effect of Casimir–Polder forces to a vertical Bose–Einstein condensate trapped in an optical lattice.

2008 - Effective mass approximation with non-parabolic bands [Articolo su rivista]
Sacchetti, Andrea
abstract

In this work we discuss the effective-mass approximation with a rapidly varying exterior potential and non-parabolic bands, recently proposed by Go´mez-Campos et al. In particular, we numerically test such an approximation confirming its validity on an explicitely solvable model.

2007 - Exponential times in the one-dimensional gross-pitaevskii equation with multiple well potential [Articolo su rivista]
Bambusi, D; Sacchetti, Andrea
abstract

We consider the Gross-Pitaevskii equation in 1 space dimension with a N-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest N eigenvalues of the linear operator is slightlydeformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected theeffective equation onMis a perturbation of a discrete nonlinear Schrödinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling amongthe wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more preciseresult showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

2007 - Resonances in twisted quantum waveguides [Articolo su rivista]
Kovarik, H; Sacchetti, Andrea
abstract

In this paper we consider embedded eigenvalues of a Schroedinger Hamiltonian in a waveguide induced by a symmetric perturbation. It is shown that these eigenvalues become unstable and turn into resonances after twisting of the waveguide. The perturbative expansion of the resonance width is calculatedfor weakly twisted waveguides and the influence of the twist on resonances in a concrete model is discussed in detail.

2007 - Second School and Workshop on "Mathematical Methods in Quantum Mechanics" [Esposizione]
Sacchetti, Andrea
abstract

Aim and topicsThe aim of the meeting is to present the state of the art in some challenging open problems in Quantum Mechanics from the point of view of Mathematical Physics. It is mainly addressed to young people interested in working on the subject. Among the topics covered: Derivation of macroscopic equations from microscopic quantum dynamics, coupled dynamics of particles and radiation fields, quantum information and entanglement, classical behaviour in quantum systems, scattering and spectral analysis for Schrödinger operators, quantum graphs.Three short courses will be given in a series of lectures scheduled in the morning of each day. Some invited talks will be given in the afternoon followed by short contributed talks given by participants.

2007 - Spectral splitting method for nonlinear Schrodinger equations with singular potential [Articolo su rivista]
Sacchetti, Andrea
abstract

We consider the time-dependent one-dimensional nonlinear Schro¨dinger equation with pointwise singular potential. Bymeans of spectral splitting methods we prove that the evolution operator is approximated by the Lie evolution operator,where the kernel of the Lie evolution operator is explicitly written. This result yields a numerical procedure which is muchless computationally expensive than multi-grid methods previously used. Furthermore, we apply the Lie approximation inorder to make some numerical experiments concerning the splitting of a soliton, interaction among solitons and blow-upphenomenon.

2007 - Stability of spectral eigenspaces in nonlinear Schrodinger equations [Articolo su rivista]
Bambusi, D; Sacchetti, Andrea
abstract

We consider the time-dependent non linear Schrodinger equationswith a double well potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time.

2006 - Stationary states for non linear one-dimensional Schrodinger equations with singular potential [Articolo su rivista]
F. F. G., Della Casa; Sacchetti, Andrea
abstract

In this paper we consider the time-independent one-dimensional non linear Schrodinger equation (NLS) with pointwise singular potential. We prove that when the strength of the pointwise interaction is less than a critical value, depending on the nonlinearity power a, then a non linear real-valued bound state exists. Furthermore, we show that when or is larger than 2 a further new real-valued stationary state appears under some conditions.

2005 - Dynamical localization for two-level systems periodically driven [Articolo su rivista]
D., Lodi; Maioli, Marco; Sacchetti, Andrea
abstract

Here, we consider a two-level system driven by an external periodic field. We show that the coherent destruction of tunnelling, as proved by Grossmann and co-workers (1991 Phys. Rev. Lett. 67 516; 1992 Europhys. Lett. 18 571) in the case of a monochromatic field, also appears for any periodic driving field given by an even regular function with zero mean value and satisfying a technical condition on the zeros of this function.

2005 - First School and Workshop "Mathematical Methods in Quantum Mechanics" [Esposizione]
Sacchetti, Andrea
abstract

The aim of the meeting is to present the state of the art in some challenging open problems in Quantum Mechanics from the point of view of Mathematical Physics. It is mainly addressed to young people interested in working on the subject.Among the topics covered: scattering for linear and nonlinear Schrödinger equation, many-body problems, derivation of macroscopic equations from quantum dynamics, Born-Oppenheimer approximation, classical behavior in quantum systems.Three short courses will be given in a series of lectures scheduled in the morning of each day. Some invited talks will be given in the afternoon followed by short contributed talks given by participants.

2005 - Nonlinear double-well Schrodinger equations in the semiclassical limit [Articolo su rivista]
Sacchetti, Andrea
abstract

We consider time-dependent Schrodinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2.

2005 - The transition from diffusion to blow-up for a nonlinear Schrodinger equation in dimension 1 [Articolo su rivista]
R., Adami; Sacchetti, Andrea
abstract

We consider the time-dependent one-dimensional nonlinear Schrodinger equation with a pointwise singular potential. We prove that if the strength of the nonlinear term is small enough, then the solution is well defined for any time, regardless of the choice of initial data; in contrast, if the nonlinearity power is larger than a critical value, for some initial data a blow-up phenomenon occurs in finite time. In particular, if the system is initially prepared in the ground state of the linear part of the Hamiltonian, then we obtain an explicit condition on the parameters for the occurrence of the blow-up.

2005 - Two level systems driven by a stochastic perturbation [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

Here we consider a two level system driven by an external harmonic field whose amplitude is perturbed by a white noise term. In the limit of small splitting, dynamical localization, i.e. coherent destruction of tunneling, is proved for times of the order of 1/epsilon, where epsilon is the two-level splitting. The same type of localization is proved if the driving field is simply the white noise.

2004 - Critical conditions for a stable molecular structure [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

Here, we show how the molecular structure appears and becomes stable for supercritical physical conditions. In particular we consider, for ammonia-type molecules, a simplified model based on a standard non-linear double-well Schrodinger equation with a dissipative term and a perturbative term representing weak collisions.

2004 - Gevrey formal power series of Wannier-Stark ladders [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

We consider time-independent Schrodinger equations in one dimension with both periodic and Stark potentials. By means of an iterative procedure we obtain a formal power series for the Wannier-Stark ladders. In the case of strongly singular periodic potentials we prove that such a formal power series is of Gevrey type.

2004 - Nonlinear time-dependent Schrodinger equations with double-well potentials [Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract

2004 - Nonlinear time-dependent Schrodinger equations: the Gross-Pitaevskii equation with double-well potential [Articolo su rivista]
Sacchetti, Andrea
abstract

We consider a class of Schrodinger equations with a symmetric double-well potential and an external, both repulsive and attractive, nonlinear perturbation. We show that, under certain conditions and in the limit of large barrier between the two wells, the reduction of the time-dependent equation to a two-mode equation gives the dominant term of the solution with a precise estimate of the error.

2004 - Nonlinear time-dependent one-dimensional Schrodinger equation with double-well potential [Articolo su rivista]
Sacchetti, Andrea
abstract

We consider time-dependent Schrodinger equations in one dimension with double-well potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest eigenvalues, then, in the semiclassical limit, we show that the reduction of the time-dependent equation to a 2-mode equation gives the dominant term of the solution with a precise estimate of the error. By means of this stability result we are able to prove the absence of the beating motion for large enough nonlinearity.

2002 - Destruction of the beating effect for a non-linear Schrodinger equation [Articolo su rivista]
V., Grecchi; A., Martinez; Sacchetti, Andrea
abstract

We consider a non-linear perturbation of a symmetric double-well potential as a model for molecular localization. In the semiclassical limit, we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect. This value is twice the one corresponding to the first bifurcation of the fundamental state. Here we make use of a particular projection operator introduced by G. Nenciu in order to extend to an infinite dimensional space some known results for a two-level system.

2002 - Instability of the tunneling destruction effect in a quasi-periodically driven two-level system [Articolo su rivista]
Sacchetti, Andrea
abstract

Here we consider the dynamics of a two-level system under an external time-dependent field. We show that in the case of a bichromatic field the dynamical localization effect is strongly sensitive with respect to the commensurability of the driving frequencies

2002 - Tunneling destruction for a nonlinear Schrodinger equation [Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract

2001 - Acceleration theorem for Bloch oscillators [Articolo su rivista]
Grecchi, V.; Sacchetti, Andrea
abstract

In this paper, we give the Heisenberg position operator in the crystal momentum representation and weprove the acceleration theorem for Bloch oscillators. As an application, we discuss the motion of well localized states.

2001 - Critical metastability and destruction of the splitting in non-autonomous systems [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

We study a periodically driven double well model. As in the case of autonomous models. previously treated in a joint paper with A. Martinez, ((7)) we have the destruction of the splitting for critical metastability. The relevance of the model for the understanding of the red shift in the inversion line of the molecule of ammonia is shortly discussed. We show that, in order to have a reasonable behavior of the metastability as a function of the frequency, a non-monochromatic perturbation is needed.

2001 - Destruction of the beating effect in a periodically driven double-well [Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract

2001 - Dynamical localization criterion for driven two-level systems [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper we consider a two-level system under the effect of an external time-dependent field. We give a precise criterion for dynamical localization. We then apply our result to the cases of external ac-dc and bichromatic field.

2000 - Molecular localization induced by collisions [Articolo su rivista]
Grecchi, V.; Sacchetti, Andrea
abstract

We consider a periodically driven double well as a simplified dynamical model for molecular localizationinduced by collisions. If the frequency of the collisions is high enough, so that the instability of the states islarger than a critical value, then the states are localized and we have the redshift of the inversion line.

1999 - First-kind Fredholm integral equations with kernel of Hankel type [Articolo su rivista]
A., Losi; Sacchetti, Andrea
abstract

We consider the first-kind Fredholm integral equatlon (A upsilon)(x) = f(x), x is an element of R+, where A is the Stieltjes transform defined as [GRAPHICS] Under some regularity assumptions on f we prove that the above problem is well-posed according to Tikhonov; that is, for any f in a given class of data there exists a unique solution upsilon of the above equation, and if \f(x)\ less than or equal to epsilon, For All x is an element of R+, for some positive is an element of then \v(y)\ less than or equal to alpha(epsilon), For All y is an element of [a, b], where alpha(epsilon) is a continuous non-decreasing function with alpha(0) = 0. An expression of the solution upsilon by means of a convergent Fourier series is also given.

1998 - Absence of the absolutely continuous spectrum for Stark-Bloch operators with strongly singular periodic potentials [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

We correct here the proof of the boundedness of the coupling term X given by us in a previous paper (1995 J. Phys. A: Math. Gen. 28 1101-6).

1998 - Wannier-Bloch oscillators [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

We consider a Wannier-Stark problem with only one ladder for weak field. We prove that a generic first-band state is a metastable state (Wannier-Bloch oscillator) oscillating because of a beating effect and decaying at the rate given by the imaginary part of the Wannier-Stark resonances. By this result we have at the same time the realization of the ideas of Bloch about the oscillations, of Wannier about the approximate quantization and of Zener about the metastability. Such oscillators, which generically perform a breathing mode motion in a large spatial region, have been experimentally observed.

1997 - Band functions for the Lame equation [Articolo su rivista]
Sacchetti, Andrea
abstract

In this paper the inversion of the elliptic Jacobian Zeta function is performed. In such a way the band functions for the Lame equation are given. A short physical and mathematical background is presented.

1997 - Lifetime of the Wannier-Stark resonances and perturbation theory [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

We consider the small field asymptotics of the lifetime of metastable states in Wannier-Stark problems. Assuming that at zero field we have Bloch operators with only the first gap open and using the regular perturbation theory, we prove that the behavior of the lifetime computed by means of the Fermi Golden Rule is proportional to the correct one with the factor (pi/3)(2). The connection with adiabatic problems is briefly discussed.

1997 - Metastable bloch oscillators [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

We give in a rigorous way the time behavior of the metastable Bloch oscillators for weak electric field. The validity of the Fermi golden rule, with the change of the numerical prefactor suggested by Kane and Blount, is definitely proved. Moreover, we give a new version of the acceleration theorem and the behavior of the Bloch oscillators in the adiabatic limit.

1996 - Double well Stark effect: Crossing and anticrossing of resonances [Articolo su rivista]
V., Grecchi; A., Martinez; Sacchetti, Andrea
abstract

We consider the semiclassical Stark effect for a family of asymmetric unstable double well models and we study the crossing and anticrossing of the field dependent resonances in the complex field plane. We prove that a Bender-Wu type singularity crosses the real axis when the internal barrier is nearly twice ''larger'' than the external one and the beating period is close to the shorter life-time of the resonances. At this critical point we have the anticrossing-crossing transition and for larger instability we have the single well localization.

1996 - Splitting instability: the unstable double wells [Articolo su rivista]
V., Grecchi; A., Martinez; Sacchetti, Andrea
abstract

In this paper we perform the semiclassical analysis of a pair of resonances in the case of a quasi-symmetrical unstable double well. We consider two kinds of asymmetric perturbations: one supported in the infinite external well, the other one of the Stark kind. We prove that the first perturbation is able to localize each state inside one of the internal wells so that we have linear Stark effect and vanishing of the splitting at the crossing point of the two resonances. This phenomenon is critical in the ratio between the internal and external barrier lengths, and the critical value of the ratio is close to two. Possible applications to the molecular structure and to the vanishing of the inversion frequency are briefly discussed.

1995 - Absence of the absolutely continuous spectrum for Stark-Bloch operators with strongly singular periodic potentials [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

We prove the absence of the absolutely continuous spectrum for the operator -d(2)/dx(2) + Sigma(j epsilon Z)alpha delta'(x - j) + fx, > 0 and alpha not equal 0, by means of the crystal momentum representation and the Howland's criterion for Floquet-type operators.

1995 - Crossing and anticrossing of resonances: the Wannier-Stark ladders [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

In the framework of regular perturbation theory we discuss the weak field crossing behavior of the resonances in double ladder (and double well) Stark problems. We get a precise condition for the anticrossing in terms of the Agmon length of the Zener barriers. This condition has a simple physical meaning: as a general rule we have anticrossing and beating effect if the lifetime of the system is larger than the beating period. Of course, we have full delocalization in the anticrossing case only.

1994 - Asymptotics of Zener double-well splittings and magnetic gaps [Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract

We consider a Zener double-well problem related to the magnetic bands in a superlattice Bloch operator. We give the precise asymptotic behaviour of the level splittings. This way we extend the Peierls substitution rule to an exponentially small term and furthermore, for the first time, we rigorously compute an exponentially small term in a Zener problem.

1994 - STARK LADDERS OF RESONANCES - WANNIER LADDERS AND PERTURBATION-THEORY [Articolo su rivista]
V., Grecchi; Maioli, Marco; Sacchetti, Andrea
abstract

Let H-B be any fixed one-dimensional Bloch Hamiltonian with only the first m gaps open and H-F = H-B + Fx be the corresponding Stark Hamiltonian. For any positive F small enough H-F has only m ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theory. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.

1994 - Stark ladders and perturbation theory [Relazione in Atti di Convegno]
Grecchi, V.; Maioli, M.; Sacchetti, Andrea
abstract

N/A

1993 - Singular continuous spectrum in a class of random Schroedinger operators [Articolo su rivista]
M., Barbieri; Maioli, Marco; Sacchetti, Andrea
abstract

For a class of random Schrodinger operators in L2(R(d)) H(omega) = -DELTA + SIGMA(j is-an-element-of Z(d)) q(j)(omega) f(x - j) where q(j) are continuous independent identically distributed bounded random variables and f has a power decay and defined sign, in any energy interval the singular continuous spectrum is either empty or with positive Lebesgue measure. As a consequence, the proof of localization for a class of random but deterministic one-dimensional operators is shifted to showing that the singular continuous spectrum has null Lebesgue measure.

1993 - Wannier ladders and perturbation theory [Articolo su rivista]
Grecchi, V; Maioli, Marco; Sacchetti, Andrea
abstract

Following Avron we consider the Stark effect for Bloch electrons in the case of a finite number of gaps. We prove that the ladders of resonances are given by the Wannier decoupled-band approximation and the perturbation theory. The Fermi golden rule yields the width behaviour of Buslaev and Dmitrieva.

1992 - Asymptotic expansion of the Stark-Wannier states [Articolo su rivista]
Sacchetti, Andrea
abstract

In this article I give an iteractive scheme to compute the coefficients of the power series expansion in the electric field parameter of the Stark-Wannier states in one-dimensional crystals. For symmetric crystals the asymptotic expansion up to the fourth order is explicitly computed, and for a solvable model the method is verified up to any order.

1992 - Stark resonances in disordered systems [Articolo su rivista]
V., Grecchi; Maioli, Marco; Sacchetti, Andrea
abstract

By slightly restricting the conditions given by Herbst and Howland, we prove the existence of resonances in the Stark effect of disordered systems (and atomic crystals) for large atomic mean distance. In the crystal case the ladders of resonances have the Wannier behavior for small complex field.

1992 - Strong asymptotic expansion for the exponential anharmonic oscillator [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

1991 - Horn of singularities for the Stark-Wannier ladders [Articolo su rivista]
V., Grecchi; Maioli, Marco; Sacchetti, Andrea
abstract

We prove that the small field asymptotic behaviour of the Stark-Wannier ladders near the real direction is generically highly singular. This result is in agreement with the conjecture of a chaotic behaviour of the lifetime of the states because of infinitely many crossings.

1991 - One-dimensional many point interactions and stability of eigenvalues [Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract

n/a

1989 - Weak-field magnetic bands in superlattices and the single-band approximation [Articolo su rivista]
Grecchi, V.; Sacchetti, Andrea
abstract

We prove the existence and we give the semiclassical magnetic asymptotics of the magnetic bands in superlattices. We use the Wannier single-band approximation which leads to a dual semiclassical Bloch model with a band function as potential. A picture of x-dependent bands suggests exponentially small magnetic gap widths as given by the beating effect of a Zener double well.

1988 - Analyticity and asymptotics for the Stark-Wannier states [Articolo su rivista]
Bentosela, F.; Caliceti, E.; Grecchi, V.; Maioli, Marco; Sacchetti, Andrea
abstract

It is proved that the Stark-Wannier states, as functions of the electric field, are analytic in a disc tangential to the real axis at the origin, with asymptotic expansion to the second order which coincides with the Wannier approximation up to the first order.