Semiclassical analysis of systems of Schrödinger equations

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for xb. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Turbulence models in the atmospheric boundary layer under convective conditions

In this work a modelization of the turbulence in the atmospheric boundary layer, under convective condition, is made. For this aim, the equations that describe the atmospheric motion are expressed through Reynolds averages and, then, they need closures. This work consists in modifying the TKE-l closure used in the BOLAM (Bologna Limited Area Model) forecast model. In particular, the single column model extracted from BOLAM is used, which is modified to obtain other three different closure schemes: a non-local term is added to the flux- gradient relations used to close the second order moments present in the evolution equation of the turbulent kinetic energy, so that the flux-gradient relations become more suitable for simulating an unstable boundary layer. Furthermore, a comparison among the results obtained from the single column model, the ones obtained from the three new schemes and the observations provided by the known case in literature ”GABLS2” is made.

Dynamical models in neuroscience: the delay FitzHugh-Nagumo equation

Il primo modello matematico in grado di descrivere il prototipo di un sistema eccitabile assimilabile ad un neurone fu sviluppato da R. FitzHugh e J. Nagumo nel 1961. Tale modello, per quanto schematico, rappresenta un importante punto di partenza per la ricerca nell'ambito neuroscientifico delle dinamiche neuronali, ed è infatti capostipite di una serie di lavori che hanno puntato a migliorare l’accuratezza e la predicibilità dei modelli matematici per le scienze. L’elevato grado di complessità nello studio dei neuroni e delle dinamiche inter-neuronali comporta, tuttavia, che molte delle caratteristiche e delle potenzialità dell’ambito non siano ancora state comprese appieno. In questo lavoro verrà approfondito un modello ispirato al lavoro originale di FitzHugh e Nagumo. Tale modello presenta l’introduzione di un termine di self-coupling con ritardo temporale nel sistema di equazioni differenziali, diventa dunque rappresentativo di modelli di campo medio in grado di descrivere gli stati macroscopici di un ensemble di neuroni. L'introduzione del ritardo è funzionale ad una descrizione più realistica dei sistemi neuronali, e produce una dinamica più ricca e complessa rispetto a quella presente nella versione originale del modello. Sarà mostrata l'esistenza di una soluzione a ciclo limite nel modello che comprende il termine di ritardo temporale, ove tale soluzione non può essere interpretata nell’ambito delle biforcazioni di Hopf. Allo scopo di esplorare alcune delle caratteristiche basilari della modellizzazione del neurone, verrà principalmente utilizzata l’impostazione della teoria dei sistemi dinamici, integrando dove necessario con alcune nozioni provenienti dall’ambito fisiologico. In conclusione sarà riportata una sezione di approfondimento sulla integrazione numerica delle equazioni differenziali con ritardo.

Geometric properties of 2-dimensional minimal surfaces in a sub-Riemannian manifold which models the Visual Cortex

In this paper we study the notion of degree forsubmanifolds embedded in an equiregular sub-Riemannian manifold and we provide the definition of their associated area functional. In this setting we prove that the Hausdorff dimension of a submanifold coincides with its degree, as stated by Gromov. Using these general definitions we compute the first variation for surfaces embedded in low dimensional manifolds and we obtain the partial differential equation associated to minimal surfaces. These minimal surfaces have several applications in the neurogeometry of the visual cortex.

Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Multi-scalar critical theories: first steps for a non perturbative functional renormalization group analysis

In this work the fundamental ideas to study properties of QFTs with the functional Renormalization Group are presented and some examples illustrated. First the Wetterich equation for the effective average action and its flow in the local potential approximation (LPA) for a single scalar field is derived. This case is considered to illustrate some techniques used to solve the RG fixed point equation and study the properties of the critical theories in D dimensions. In particular the shooting methods for the ODE equation for the fixed point potential as well as the approach which studies a polynomial truncation with a finite number of couplings, which is convenient to study the critical exponents. We then study novel cases related to multi field scalar theories, deriving the flow equations for the LPA truncation, both without assuming any global symmetry and also specialising to cases with a given symmetry, using truncations based on polynomials of the symmetry invariants. This is used to study possible non perturbative solutions of critical theories which are extensions of known perturbative results, obtained in the epsilon expansion below the upper critical dimension.

A Comparative Analysis of Dynamical Models for Tidal Dissipation in Natural Satellite Ephemerides

The study of the tides of a celestial bodies can unveil important information about their interior as well as their orbital evolution. The most important tidal parameter is the Love number, which defines the deformation of the gravity field due to an external perturbing body. Tidal dissipation is very important because it drives the secular orbital evolution of the natural satellites, which is even more important in the case of the the Jupiter system, where three of the Galilean moons, Io, Europa and Ganymede, are locked in an orbital resonance where the ratio of their mean motions is 4:2:1. This is called Laplace resonance. Tidal dissipation is described by the dissipation ratio k2/Q, where Q is the quality factor and it describes the dampening of a system. The goal of this thesis is to analyze and compare the two main tidal dynamical models, Mignard's model and gravity field variation model, to understand the differences between each model with a main focus on the single-moon case with Io, which can help also understanding better the differences between the two models without over complicating the dynamical model. In this work we have verified and validated both models, we have compared them and pinpointed the main differences and features that characterize each model. Mignard's model treats the tides directly as a force, while the gravity field variation model describes the tides with a change of the spherical harmonic coefficients. Finally, we have also briefly analyzed the difference between the single-moon case and the two-moon case, and we have confirmed that the governing equations that describe the change of semi-major axis and eccentricity are not good anymore when more moons are present.