Lorenzo Zanelli

GEOMETRIC APPROACH TO THE HAMILTON–JACOBI EQUATION AND GLOBAL PARAMETRICES FOR THE SCHRÖDINGER PROPAGATOR

We construct a family of global Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrodinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton–Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

Schrödinger spectra and the effective Hamiltonian of weak KAM theory on the flat torus

In this paper we investigate the link between the spectrum of some periodic Schrodinger type operators and the effective Hamiltonian of the weak KAM theory. We show that the extension of some local quasimodes is linked to the localization of the Schrodinger spectrum. Such a result provides additional information with respect to the well known Bohr-Sommerfeld quantization rules, here in a more general setting than the integrable or quasi-integrable ones.

Global parametrices for the Schrödinger propagator and geometric approach to the Hamilton-Jacobi equation

— A result is announced concerning a family of semiclassical Fourier Integral Operators representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. The construction is based on the geometrical approach of the corresponding HamiltonJacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

A PDE APPROACH TO FINITE TIME INDICATORS IN ERGODIC THEORY

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition — that we show robust under suitable perturbations — allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.

Integral representations of the schrödinger propagator

We consider the Schrodinger equation for the Hamiltonian operator H = −ħ2/2m + Δ + V (x), where V is a potential function modeling one-particle scattering problems. By means of a strongly converging regularization of the Schrodinger propagator U(t), we introduce a new class of integral representations for the relaxed kernel in terms of oscillatory integrals. They are constructed with complex amplitudes and real phase functions that belong to the class of global weakly quadratic generating functions of the Lagrangian submanifolds Λt ⊂ T★ℝn × T★ℝn related to the group of classical canonical transformations olH. Moreover, as a particular generating function, we consider the action functional A[γ] = ∫0t ½ m ׀ẏ(s)׀2 − V(γ y(s))ds evaluated on a suitable finite-dimensional space of curves γ ∈ Γ ⊂ H1 ([0, t],ℝn). As a matter of fact we obtain a finite-dimensional path integral representation for the relaxed kernel.

Mechanical properties variation and constitutive modelling of biomedical polymers after sterilization

PURPOSE In this work, the mechanical behavior of two block copolymers for biomedical applications is studied with particular regard to the effects induced by a steam sterilization treatment that biomedical devices usually undergo in healthcare facilities. This investigation is aimed at describing the elasto-plastic behavior of the stress-strain response, determining a functional dependence between material constitutive parameters, to obtain an optimal constitutive model. METHODS The mechanical properties of these polymers are analyzed through uniaxial tensile tests, before and after the sterilization process. The effect of sterilization on the mechanical behavior is evaluated. The Ramberg-Osgood model is used to describe the elasto-plastic behavior of the stress-strain response. RESULTS Data from uniaxial tensile tests are discussed in the light of previous data on the same polymeric materials, in order to highlight the correlation between physicochemical and mechanical properties variation. The material constitutive parameters are determined and the functional dependence between them is found, thus enabling an optimal constitutive model to be obtained. CONCLUSIONS The effect of sterilization on the material constitutive parameters is studied, to evaluate the suitability of the model in describing the mechanical response of biomedical polymer before and after sterilization treatment. The same approach can be applied to other biomaterials, under various tensile tests, and for several processes that induce variation in mechanical properties.