arXiv subjects

Carlos Valero

Publications and source records attributed to Carlos Valero.

11 recordsLinked to original sources

arXiv2025

A Calder\'on Problem for the Dirac operator with chiral boundary conditions

We consider on a spin manifold with boundary a Dirac operator $D_A$ with chiral boundary conditions, twisted by a unitary connection $A$. When $m$ is not in the chiral spectrum of $D_A$, we define an analogue of the Dirichlet-to-Neumann map for the Dirac equation $D_A - m$, which we call the boundary conjugation map, and show that it is a pseudodifferential operator of order $0$ on the boundary. We show that in dimension greater than 2, its symbol determines the Taylor series of the metric and connection modulo gauge on the boundary when $m \neq 0$ and $m^2$ is not in the Dirichlet spectrum of $D_A^2$. We go on to show that a real-analytic Riemannian manifold and twisted spinor bundle with twisted spin connection can be recovered from its boundary conjugation map. Under further hypotheses, one can recover the unitary connection up to global gauge equivalence and the complex spinor bundles. Similar results hold in dimension $2$ when the auxiliary bundle and connection are absent.

math.AP↗

arXiv2024

A Calder\'{o}n Problem for Beltrami Fields

On a $3$-dimensional Riemannian manifold with boundary, we define an analogue of the Dirichlet-to-Neumann map for Beltrami fields, which are the eigenvectors of the curl operator and play a major role in fluid mechanics. This map sends the normal component of a Beltrami field to its tangential component on the boundary. In this paper we establish two results showing how this normal-to-tangential map encodes geometric information on the underlying manifold. First, we show that the normal-to-tangential map is a pseudodifferential operator of order zero on the boundary whose total symbol determines the Taylor series of the metric at the boundary. Second, we go on to show that a real-analytic simply connected $3$-manifold can be reconstructed from its normal-to-tangential map. Interestingly, since Green's functions do not exist for the Beltrami field equation, a key idea of the proof is to find an appropriate substitute, which turn out to have a natural physical interpretation as the magnetic fields generated by small current loops.

math.AP↗

arXiv2022

A Uniqueness Result for the Calderon Problem for $U(N)$-connections coupled to spinors

In this paper we define a Dirichlet-to-Neumann map for a twisted Dirac Laplacian acting on bundle-valued spinors over a spin manifold. We show that this map is a pseudodifferential operator of order 1 whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if two real-analytic connections couple to a spinor via the Yang--Mills--Dirac equations with appropriate boundary conditions, and have equal Dirichlet-to-Neumann maps, then the two connections are locally gauge equivalent. In the abelian case, the connections are globally gauge equivalent.

math.AP↗

arXiv2022

Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space

We review a new theory of orthogonal separation of variables on pseudo-Riemannian spaces of constant zero curvature via concircular tensors and warped products. We then apply this theory to three-dimensional Minkowski space, obtaining an invariant classification of the forty-five orthogonal separable webs modulo the action of the isometry group. The eighty-eight inequivalent coordinate charts adapted to the webs are also determined and listed. We find a number of separable webs which do not appear in previous works in the literature. Further, the method used seems to be more efficient and concise than those employed in earlier works.

math-ph↗

arXiv2020

A Coordinate Free Formulation of Effective Diffusion on Channels

We study diffusion processes in regions generated by sliding a cross section by the phase flow of vector filed on curved spaces of arbitrary dimension. We do this by studying the effective diffusion coefficient D that arises when trying to reduce the n-dimensional diffusion equation to a 1-dimensional diffusion equation by means of a projection method. We use the mathematical language of exterior calculus to derive a coordinate free formula for this coefficient in both infinite and finite transversal diffusion rate cases. The use of these techniques leads to a formula for D which provides a deeper understanding of effective diffusion than when using a coordinate dependent approach.

math-ph↗

arXiv2020

First Order Symbols on Surfaces and their multiplicity sets

We study the multiplicity sets of first order symbols associated with differential operators on two dimensional surfaces. This work is inspired by the phenomenon of conical refraction explained by the existence of singularities in the Fresnel hyper-surface for Maxwell's equations on an anisotropic crystal.

math-ph↗

arXiv2019

Singularities of the eigenvalue functions for first order symmetric symbols on rank two vector bundles over surfaces

For a rank two bundle $F$ over a surface $X$, we study the set of singularities $M_σ$ of the eigenvalue functions of symmetric symbols $σ$ associated to first order differential operators on $F$. We prove that the existence of these singularities follows from topological considerations. We define the degree of the set of singularities and show that it can be used to count the number of directions at which special optical phenomena occur. For the case when $F=TX$ we compute a formula for the degree in terms of the Euler characteristics of $X$ and $N_σ$, where $N_σ\subset X$ is a manifold with boundary associated to $σ$.

math-ph↗

arXiv2018

Anisotropy in 2D Discrete Exterior Calculus

We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (element by element). It also allows us to deduce, in a robust manner, anisotropic fluxes and the DEC discretization of the pullback of 1-forms by the anisotropy tensor, i.e. we deduce how the anisotropy tensor acts on primal 1-forms. Due to the local formulation, the computational cost of DEC is similar to that of the Finite Element Method with Linear interpolations functions (FEML). The numerical DEC solutions to the anisotropic Poisson equation show numerical convergence, are very close to those of FEML on fine meshes and are slightly better than those of FEML on coarse meshes.

math.NA↗

arXiv2018

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional Hyperbolic and de Sitter spaces

We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for Hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

math-ph↗

arXiv2016

Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.

math-ph↗

arXiv2013

Maxwell's Equations, The Euler Index and Morse Theory

We show show that the singularities of the Fresnel surface for Maxwell's equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to de-singularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.

math-ph↗