# Trabajos en física teórica o matemáticas

Nodal set of monochromatic waves satisfying the Random Wave Model.

We construct deterministic solutions to the Helmholtz equation in $\mathbb{R}^m$ which behave accordingly to the Random Wave Model. We then find the number of their nodal domains, their nodal volume and the topologies and nesting trees of their nodal set in growing balls around the origin. The proof of the pseudo-random behaviour of the functions under consideration hinges on a de-randomisation technique pioneered by Bourgain and proceeds via computing their $L^p$-norms. The study of their nodal set relies on its stability properties and on the evaluation of their doubling index, in an average sense.

Beltrami fields exhibit knots and chaos almost surely

In this paper we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, hich arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on $\mathbb{R}^3$ and of high-frequency Beltrami fields on the 3-torus.

We study monochromatic random waves on $\mathbb{R}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves, showing that the number of nodal components contained in a large ball $B_R$ grows asymptotically like $R/\pi$ with probability $p_n>0$ and is bounded uniformly in $R$ with probability $1-p_n$ (which is positive if and only if $n\ge 3$⁠). In the latter case, we show the existence of a unique noncompact nodal component. We also provide an explicit sufficient stability criterion to ascertain when a more general Gaussian probability distribution has the same asymptotic nodal distribution law.

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.

Resumen:

En este trabajo se abordan los sistemas superintegrables tanto en mecánica clásica como cuántica. Primero, y con cierta generalidad, se estudia una formulación geométrica de la mecánica. Estudiamos esta formulación moderna de la mecánica con teoría de variedades y cálculo tensorial. Debido a la complejidad/abstracción matemática inicial, intentamos dar una visión heurística de los resultados más básicos o importantes. Después, se realiza una introducción teórica a la superintegrabilidad, seguida por ejemplos importantes (Kepler y Coulomb en $\mathbb{E}^3$ y $\mathbb{S}^2$) estudiados haciendo uso de sus propiedades como sistemas superintegrables.
En la segunda parte del trabajo, realizamos un estudio de una nueva familia de hamiltonianos en $\mathbb{S}^2$. Desarrollamos tanto un análisis del caso clásico como del cuántico. Utilizamos el formalismo de factorizaciones, operadores escalera (ladder) y desplazamiento (shift). Este estudio, consta de dos partes. Por un lado, proponemos una serie de resultados nuevos (con su correspondiente demostración) lo más generales posibles para que sean aplicables a otros sistemas físicos. Por otro lado, aplicamos esos resultados a nuestro sistema particular (hamiltoniano U(3) generalizado) siguiendo las ideas de lo sistemas TTW y mostramos sus propiedades asociadas a la superintegrabilidad.