
Andrea SACCHETTI
Professore Ordinario Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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Pubblicazioni
2023
 Enhancement of the Zakharov–Glassey’s method for Blowup in nonlinear Schrödinger equations
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper we give a sharper sufficient condition for blowup 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 Lietype spectral splitting approximation where the
external potential is of quadratic type. It is proved that we can approximate the solution
to a onedimensional 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 TightBinding Approximation for TimeDependent Nonlinear Schrödinger Equations
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper, we consider the nonlinear onedimensional 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 tightbinding 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 doublewell Bose–Einstein condensate
[Articolo su rivista]
Gavioli, A.; Sacchetti, A.
abstract
In this paper we consider a twomode dynamical system as a model for a driven onedimensional damped Bose–Einstein condensate in a doublewell 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 timedependent 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 quasiperiodic 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 quasiperiodic 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 onedimensional Schrodinger equation with a periodic potential and a Starktype 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 StarkWannier Equation
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper we consider stationary solutions to the nonlinear onedimensional Schrödinger equation with a periodic potential and a Starktype perturbation. In the limit of large periodic potential the StarkWannier 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 StarkWannier ladders for accelerated BoseEinstein 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 StarkWannier 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 BoseâEinstein 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 nearestneighbor 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
 Doublebarrier 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 onedimensional 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 doublebarrier 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 BoseEinstein condensates in a doublewell potential
[Articolo su rivista]
Sacchetti, Andrea
abstract
Devices based on ultracold atoms moving in an accelerating optical lattice or doublewell 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 phaseshift, 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 multiplewell potential and a Starktype perturbation
[Articolo su rivista]
Sacchetti, Andrea
abstract
A BoseEinstein condensate (BEC) confined in a onedimensional lattice under the effect of an external homogeneous field is described by the GrossPitaevskii 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 finitedimensional 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 doublebarrier model
[Articolo su rivista]
Sacchetti, Andrea
abstract
Here we consider the time evolution of a onedimensional 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,eitH φ〉, where H is the doublebarrier Hamiltonian operator and where ψ and f are two test functions.
2015
 Solution to the doublewell nonlinear Schrödinger equation with Starktype external field
[Articolo su rivista]
Sacchetti, Andrea
abstract
Here we consider one and twodimensional nonlinear Schrödinger equations with double well potential and a Starktype perturbation term. In the semiclassical 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 StarkWannier quantum resonances
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper, we prove the existence of the StarkWannier quantum resonances for
onedimensional 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 BoseEinstein
condensates in optical lattices
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper we consider BoseEinstein condensates (BECs) in one, two and threedimension
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 onsite interaction term and the hopping matrix element. Such an effect can be
directly seen in the GrossPitaevskii equation with doublewell potentials and it also explains the different
behavior between onedimensional models and two/threedimensional 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 onedimensional nonlinear Schrödinger equation
with a periodic potential. In the semiclassical limit we prove the existence of stationary
solutions by means of the reduction of the nonlinear Schrödinger equation to a discrete
nonlinear 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 onedimensional periodic lattice is also discussed
2014
 Stationary solutions to the
multidimensional Gross–Pitaevskii
equation with doublewell potential
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper we consider a nonlinear Schr ̈odinger equation with a cubic
nonlinearity and a multidimensional 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 multiplewell potential
[Articolo su rivista]
Sacchetti, Andrea
abstract
We consider the stationary solutions for a class of Schrödinger equations with a Nwell 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 Ndimensional 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 symmetrybreaking 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 doublewell potential and a nonlinear perturbation. Here, in the semiclassicallimit we prove that the reduction to a finitemode approximation give the stationary solutions,up to an exponentially small term, and that symmetrybreaking 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 onedimensional 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 openboundary Schroedinger equation, we singleout the effects of the twist and show how the presence of a localized state leads to a BreitWigner 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 travelingstates energy range can reduce theirdetrimental effects on coherent transport.
2011
 Resonant states for a threebody 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 manybody problems under an external field: the molecule ion H2 under the effect of an external Starktype 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 Starktype 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 timedependent onedimensional nonlinear Schrodinger equation with a symmetric doublewell 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 Strichartztype 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 ultrasmall 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 ultrasmall 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 BoseEinstein condensates
[Articolo su rivista]
Sacchetti, Andrea
abstract
Here, we consider damped twocomponent BoseEinstein condensates with manybody interactions. We show that, when the external trapping potential has a doublewell 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 onedimensional BoseEinstein condensates with defects
[Capitolo/Saggio]
R., Adami; D., Noja; Sacchetti, Andrea
abstract
BoseEinstein condensation and the related topic of GrossPitaevskii equation have become an important source of models and problems in mathematical physics and analysis. In particular, in the last decade, the interest in lowdimensional systems that evolve through the nonlinear Schroedinger equation has undergone an impressive growth. The reason is twofold: on the one hand, effectively onedimensional BoseEinstein condensates are currently realized, and the investigation on their dynamics isnowadays a welldeveloped field for experimentalists. On the other hand, in contrast to its higherdimensional analogous,the onedimensional 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 socalled 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 saddlenode 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 nonparabolic bands
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this work we discuss the effectivemass approximation with a rapidly varying exterior potential and nonparabolic bands, recently proposed by Go´mezCampos et al. In particular, we numerically test such an approximation confirming its validity on an explicitely solvable model.
2007
 Exponential times in the onedimensional grosspitaevskii equation with multiple well potential
[Articolo su rivista]
Bambusi, D; Sacchetti, Andrea
abstract
We consider the GrossPitaevskii equation in 1 space dimension with a Nwell 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 nonresonance 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 timedependent onedimensional 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 multigrid 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 blowupphenomenon.
2007
 Stability of spectral eigenspaces in nonlinear Schrodinger equations
[Articolo su rivista]
Bambusi, D; Sacchetti, Andrea
abstract
We consider the timedependent 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 onedimensional Schrodinger equations with singular potential
[Articolo su rivista]
F. F. G., Della Casa; Sacchetti, Andrea
abstract
In this paper we consider the timeindependent onedimensional 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 realvalued bound state exists. Furthermore, we show that when or is larger than 2 a further new realvalued stationary state appears under some conditions.
2005
 Dynamical localization for twolevel systems periodically driven
[Articolo su rivista]
D., Lodi; Maioli, Marco; Sacchetti, Andrea
abstract
Here, we consider a twolevel system driven by an external periodic field. We show that the coherent destruction of tunnelling, as proved by Grossmann and coworkers (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, manybody problems, derivation of macroscopic equations from quantum dynamics, BornOppenheimer 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 doublewell Schrodinger equations in the semiclassical limit
[Articolo su rivista]
Sacchetti, Andrea
abstract
We consider timedependent Schrodinger equations with a double well potential and an external nonlinear, both local and nonlocal, 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 blowup for a nonlinear Schrodinger equation in dimension 1
[Articolo su rivista]
R., Adami; Sacchetti, Andrea
abstract
We consider the timedependent onedimensional 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 blowup 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 blowup.
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 twolevel 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 ammoniatype molecules, a simplified model based on a standard nonlinear doublewell Schrodinger equation with a dissipative term and a perturbative term representing weak collisions.
2004
 Gevrey formal power series of WannierStark ladders
[Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract
We consider timeindependent 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 WannierStark ladders. In the case of strongly singular periodic potentials we prove that such a formal power series is of Gevrey type.
2004
 Nonlinear timedependent Schrodinger equations with doublewell potentials
[Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract
2004
 Nonlinear timedependent Schrodinger equations: the GrossPitaevskii equation with doublewell potential
[Articolo su rivista]
Sacchetti, Andrea
abstract
We consider a class of Schrodinger equations with a symmetric doublewell 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 timedependent equation to a twomode equation gives the dominant term of the solution with a precise estimate of the error.
2004
 Nonlinear timedependent onedimensional Schrodinger equation with doublewell potential
[Articolo su rivista]
Sacchetti, Andrea
abstract
We consider timedependent Schrodinger equations in one dimension with doublewell 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 timedependent equation to a 2mode 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 nonlinear Schrodinger equation
[Articolo su rivista]
V., Grecchi; A., Martinez; Sacchetti, Andrea
abstract
We consider a nonlinear perturbation of a symmetric doublewell 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 twolevel system.
2002
 Instability of the tunneling destruction effect in a quasiperiodically driven twolevel system
[Articolo su rivista]
Sacchetti, Andrea
abstract
Here we consider the dynamics of a twolevel system under an external timedependent 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 nonautonomous 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 nonmonochromatic perturbation is needed.
2001
 Destruction of the beating effect in a periodically driven doublewell
[Relazione in Atti di Convegno]
Sacchetti, Andrea
abstract
2001
 Dynamical localization criterion for driven twolevel systems
[Articolo su rivista]
Sacchetti, Andrea
abstract
In this paper we consider a twolevel system under the effect of an external timedependent field. We give a precise criterion for dynamical localization. We then apply our result to the cases of external acdc 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
 Firstkind Fredholm integral equations with kernel of Hankel type
[Articolo su rivista]
A., Losi; Sacchetti, Andrea
abstract
We consider the firstkind 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 wellposed 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 nondecreasing 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 StarkBloch 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 11016).
1998
 WannierBloch oscillators
[Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract
We consider a WannierStark problem with only one ladder for weak field. We prove that a generic firstband state is a metastable state (WannierBloch oscillator) oscillating because of a beating effect and decaying at the rate given by the imaginary part of the WannierStark 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 WannierStark 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 WannierStark 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 BenderWu 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 lifetime of the resonances. At this critical point we have the anticrossingcrossing 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 quasisymmetrical 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 StarkBloch 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 Floquettype operators.
1995
 Crossing and anticrossing of resonances: the WannierStark 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 doublewell splittings and magnetic gaps
[Articolo su rivista]
V., Grecchi; Sacchetti, Andrea
abstract
We consider a Zener doublewell 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 PERTURBATIONTHEORY
[Articolo su rivista]
V., Grecchi; Maioli, Marco; Sacchetti, Andrea
abstract
Let HB be any fixed onedimensional Bloch Hamiltonian with only the first m gaps open and HF = HB + Fx be the corresponding Stark Hamiltonian. For any positive F small enough HF 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 isanelementof 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 onedimensional 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 decoupledband approximation and the perturbation theory. The Fermi golden rule yields the width behaviour of Buslaev and Dmitrieva.
1992
 Asymptotic expansion of the StarkWannier 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 StarkWannier states in onedimensional 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 StarkWannier ladders
[Articolo su rivista]
V., Grecchi; Maioli, Marco; Sacchetti, Andrea
abstract
We prove that the small field asymptotic behaviour of the StarkWannier 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
 Onedimensional many point interactions and stability of eigenvalues
[Articolo su rivista]
Maioli, Marco; Sacchetti, Andrea
abstract
n/a
1989
 Weakfield magnetic bands in superlattices and the singleband 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 singleband approximation which leads to a dual semiclassical Bloch model with a band function as potential. A picture of xdependent bands suggests exponentially small magnetic gap widths as given by the beating effect of a Zener double well.
1988
 Analyticity and asymptotics for the StarkWannier states
[Articolo su rivista]
Bentosela, F.; Caliceti, E.; Grecchi, V.; Maioli, Marco; Sacchetti, Andrea
abstract
It is proved that the StarkWannier 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.