Huygens' principle and Dirac-Weyl equation

We investigate the validity of Huygens' principle for forward propagation in the massless Dirac-Weyl equation. The principle holds for odd space dimension n, while it is invalid for even n. We explicitly solve the cases n=1,2 and 3 and discuss generic $n$. We compare with the massless Klein-Gordon equation and comment on possible generalizations and applications.Comment: 7 pages, 1 figur

Representation of non-semibounded quadratic forms and orthogonal additivity

In this article we give a representation theorem for non-semibounded Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint operator. The main assumptions are closability of the Hermitean quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail

On the theory of self-dajoint extensions of the Laplace-Beltrami operator quadratic forms and symmetry

The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to describe them. Moreover, we want to emphasise the role that quadratic forms can play in the description of quantum systems. ------------------------------------------------------------------------------------------------------------El objetivo principal de esta memoria es analizar en detalle tanto la construcción de extensiones autoadjuntas del operador de Laplace-Beltrami definido sobre una variedad Riemanniana compacta con frontera, como el papel que juegan las formas cuadráticas a la hora de describirlas. Más aún, queremos enfatizar el papel que juegan las formas cuadráticas a la hora de describir sistemas cuánticos

On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics

On a sharper bound on the stability of non-autonomous Schr\"odinger equations and applications to quantum control

We study the stability of the Schr\"odinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schr\"odinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the literature. This makes it a potentially useful tool for time-dependent problems in Quantum Physics, in particular for Quantum Control. We apply this result to prove two theorems about global approximate controllability of infinite-dimensional quantum systems. These results improve and generalise existing results on infinite-dimensional quantum control.Comment: arXiv admin note: text overlap with arXiv:2108.0049

Aspects of geodesical motion with Fisher-Rao metric: classical and quantum

The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon's Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.Comment: 15 pages, 5 figure

On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

We study two seminal approaches, developed by B. Simon and J. Kisynski, to the wellposedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterisation of the continuity and differentiability properties of form-valued and operator-valued functions, which enables an extensive comparison between the two approaches and their technical assumptions.A.B. and J.M.P.-P. acknowledge support provided by the “Ministerio de Ciencia e Innovación” Research Project PID2020-117477GB-I00, by the QUITEMAD Project P2018/TCS-4342 funded by Madrid Government (Comunidad de Madrid-Spain) and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of “Research Funds for Beatriz Galindo Fellowships” (C&QIG-BG-CM-UC3M), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation). A.B. acknowledges financial support by “Universidad Carlos III de Madrid” through Ph.D. Program Grant PIPF UC3M 01-1819 and through its mobility grant in 2020. D.L. was partially supported by “Istituto Nazionale di Fisica Nucleare” (INFN) through the project “QUANTUM” and the Italian National Group of Mathematical Physics (GNFM-INdAM)

Potential fields modeling for the Cayos Basin (Western Caribbean Plate): Implications in basin crustal structure

The Cayos Basin is an offshore basin located in the Colombian Caribbean Sea and forms part of the Lower Nicaraguan Rise, a geological province of the western region of the Caribbean Plate. Until now, the origin of the province is still being debated. Advanced research in the study area regarding its composition and structure, from land outcrops, petrology and geochemistry of drilled cores and dredged samples, and geophysical investigations, indicates a volcanic origin for this geological province, and a close relationship to the formation of the Caribbean Large Igneous Province. On the contrary, other studies suggest that the Lower Nicaraguan Rise may be part of the continental Chortis block. In this paper, we present and discuss alternative scenarios for the nature of the underlying crust below the sedimentary sequences in the Cayos Basin. We characterize the basin through the interpretation of magnetic and gravity anomaly grids, and 2D forward modeling, constructed based on three sections, by considering restriction seismic data from previous works. The results show that the Cayos Basin is underlain by geological bodies with high density and higher magnetization. From the gravity and magnetic forward modeling, we estimated the depth to the basement is about 2-6 km, and the Moho discontinuity to have an average of 18 km below, the Cayos Basin. Our investigation implies that, at least, the Cayos Basin is in the oceanic crust domain and shows no evidence of a continental source of the Chortis block