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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.

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.

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.

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 〈ψ, e −itH ϕ〉 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 | 〈ψ, e −itH ψ〉 |2 is given.

A Bose-Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross-Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrodinger equation. Then, by means of numerical experiments we show that the BEC's center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period., 24 pages, 8 figures

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., 5 figures

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.

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.

Here, we show how the molecular structure appears and becomes stable for supercritical physical conditions. In particular we consider, for the ammonia molecule in a gas, a simple model based on a standard non-linear double-well Schroedinger equation with a dissipative term and a term representing weak collisions., Comment: 4 pages, 2 figures

In this paper we consider a one-dimensional non-linear Schrodinger 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 Schrodinger equation to a discrete non-linear Schrodinger 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.

In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger 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., 30 pages, 1 figure

Here we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of two repulsive Dirac's deltas. In such a "pedagogical" model we give, by means of the theory of quantum resonances, the explicit expression of the dominant terms of $$, where $H$ is the double-barrier Hamiltonian operator and where $\psi$ and $\phi$ are two test functions., Comment: 16 pages, 3 figures

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.

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 calculated for weakly twisted waveguides and the influence of the twist on resonances in a concrete model is discussed in detail., 15 Pages

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/e, where e is the two-level splitting. The same type of localization is proved if the driving field is simply the white noise.

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

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.

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 \(\). The connection with adiabatic problems is briefly discussed.

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.

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.

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 plal}e. 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.

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.

LetHB be any fixed one-dimensional Bloch Hamiltonian with only the firstm gaps open andHF=HB+Fx be the corresponding Stark Hamiltonian. For any positiveF small enoughHF has onlym ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theroy. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.

We consider the Bloch problems with a finite number of open gaps and we prove, for any external weak enough electric field, the existence of a finite number of Stark ladders, given by complex translation, decoupled band approximation and regular perturbation theory.

Following Avron (1982) the authors consider the Stark effect for Bloch electrons in the case of a finite number of gaps. They 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 Dmitirieva (1990).

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

In this paper we discuss the splitting instability in a periodically driven double well. The physical motivation of this study comes from the relevance of the concept of molecular structure in chemistry, but the model could be tested directly by means of heterostructures and microwaves. Let us recall the old problem of the explanation of the molecular localization ~ML! hypothesis, successfully used in chemistry as the concept of molecular structure, in the rigorous quantummechanics ~QM! framework @1#. QM requires that the probability distribution of stationary states have the same symmetry of the Hamiltonian, in marked contrast with the ML requirement. The qualitative explanation of this apparent contradiction is simple: since the molecule is not an isolated system, its states cannot be stationary @2#. The main problem is the understanding of the quantitative aspect of the phenomenon, as it results from the following question by Woolley @1#: ‘‘Why should the general quantum theory describing energy eigenstates turn out to be of such little use in chemistry, or put in another way, why should transitions out of the time-dependent molecular quantum states which empirically appear to be an essential ingredient of any useful quantum chemistry, be so slow?’’ Although it is generally accepted that the phenomenon should be explained by means of decoherence arguments @3#, it is also clear that explicit models are needed. Thus, by means of the study of an explicit model, we want to point out the role of instability in the localization phenomenon. Indeed, we expect the existence of metastable states in perturbed systems and we want to study the smallness of the interaction between a pair of such states for large instability. Let us consider the case of the ammonia molecule NH3, where the model for the motion of the nitrogen atom N is a double well with a large internal barrier @1#. In this model we have the pyramidal shape of the molecule ~molecular structure! if the state is localized in one of the wells. The inversion line of the molecular microwave emission gives the energy splitting of the stationary states. Experiments on ammonia gas show that the localization and the inversion line are dependent on pressure. In particular, the localization probability increases and the inversion line broadens and decreases as the pressure increases, giving the so-called redshift ~RS! effect @2#. Some previous explicit models @4# which are able to explain ML, are autonomous, i.e., they make use of timeindependent potentials. In particular in a recent paper @5# ,b y using an unstable autonomous model, both ML and RS are obtained. In the present paper we use a nonautonomous model ~time-dependent potential! so that the instability caused by the molecular, collisions is represented in a more realistic way. In particular, our model consists of a double-well potential with a time-dependent perturbation simulating the dynamical influence of the environment on the ammonia molecule, i.e., the collisions with the other molecules of the gas, where the collision frequency is related to the pressure. Let us notice that the present model is more physical than the previous ones for the reasons stated above, but it is still simplified. One simplification is the choice of a perturbation periodic in time. This choice is technical and is due to the recent improvement of methods for handling periodic problems. We point out that the classical resonance effect between different frequencies are not relevant for the results. In any case the results give an a posteriori justification of the model. Since the first~but not the second! order perturbation term vanishes, we set the perturbation of the same order of the square root of the splitting. We consider the large internal barrier regime, so that both the splitting and the perturbation are exponentially small. This choice of parameters is similar to previous ones, and allows us to apply the same comparison with experiments given by Claverie and Jona Lasinio @4#, although in that case the frequency parameter was absent. If the periodic perturbation is strong enough, and the Fourier coefficients are slowly decreasing for increasing index, we have localization for a frequency larger than a critical value but smaller than a large value. This upper bound should be related to the simplification of the model given by the periodicity of the time behavior. Since the model is linear, we have no spontaneous symmetry breaking, so that the perturbation is asymmetric, but not too much, in order to have the RS. We control the asymmetry of the perturbation by varying the coefficient of the time independent perturbation.