arXiv subjects

Luiz C. B. da Silva

Publications and source records attributed to Luiz C. B. da Silva.

22 records · Page 2Linked to original sources

arXiv2018

Schrödinger formalism for a particle constrained to a surface in $\mathbb{R}_1^3$

In this work it is studied the Schrödinger equation for a non-relativistic particle restricted to move on a surface $S$ in a three-dimensional Minkowskian medium $\mathbb{R}_1^3$, i.e., the space $\mathbb{R}^3$ equipped with the metric $\text{diag}(-1,1,1)$. After establishing the consistency of the interpretative postulates for the new Schrödinger equation, namely the conservation of probability and the hermiticity of the new Hamiltonian built out of the Laplacian in $\mathbb{R}_1^3$, we investigate the confining potential formalism in the new effective geometry. Like in the well-known Euclidean case, it is found a geometry-induced potential acting on the dynamics $V_S = - \frac{\hbar^{2}}{2m} \left(\varepsilon H^2-K\right)$ which, besides the usual dependence on the mean ($H$) and Gaussian ($K$) curvatures of the surface, has the remarkable feature of a dependence on the signature of the induced metric of the surface: $\varepsilon= +1$ if the signature is $(-,+)$, and $\varepsilon=1$ if the signature is $(+,+)$. Applications to surfaces of revolution in $\mathbb{R}^3_1$ are examined, and we provide examples where the Schrödinger equation is exactly solvable. It is hoped that our formalism will prove useful in the modeling of novel materials such as hyperbolic metamaterials, which are characterized by a hyperbolic dispersion relation, in contrast to the usual spherical (elliptic) dispersion typically found in conventional materials.

math-ph↗

arXiv2018

Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle

This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any other orthonormal moving frame, from which one defines Rotation Minimizing (RM) frames as the ones that achieve this minimum. Remarkably, RM frames are ideal to study spherical curves and allow us to characterize them through a simple linear equation. We also apply these ideas to curves that lie on level surfaces, by reinterpreting the problem in the context of a metric induced by a Hessian, which may fail to be positive or non-degenerate and naturally leads us to a Lorentz-Minkowski $\mathbb{E}_1^3$ or isotropic $\mathbb{I}^3$ space. Here we develop a systematic approach to construct RM frames and characterize spherical curves in $\mathbb{E}_1^3$ and $\mathbb{I}^3$ and furnish a criterion for a curve to lie on a level surface. Finally, we extend these tools to characterize geodesic spherical curves in hyperbolic and spherical spaces. In the 2nd part we apply the previous ideas in the quantum dynamics of a constrained particle. After describing the confining potential formalism, from which emerges a geometry-induced potential (GIP), we devote our attention to tubular surfaces to model curved nanotubes. The use of RM frames offers a simpler description for the constrained dynamics and allows us to show that the torsion of the centerline of a tube gives rise to a geometric phase. Later, we study the problem of prescribed GIP for surfaces. Here we explore the GIP for surfaces invariant by a 1-parameter group of isometries, i.e., cylindrical, revolution, and helicoidal surfaces. Finally, we devote a special attention to helicoidal minimal surfaces and prove the existence of geometry-induced bound and localized states.

math.DG↗

arXiv2018

Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.

math.DG↗

arXiv2017

Moving frames and the characterization of curves that lie on a surface

In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples: planes, spheres, or cylinders. Generally, the characterization of such curves, both in Euclidean ($E^3$) and in Lorentz-Minkowski ($E_1^3$) spaces, involves an ODE relating curvature and torsion. However, by equipping a curve with a relatively parallel moving frame, Bishop was able to characterize spherical curves in $E^3$ through a linear equation relating the coefficients which dictate the frame motion. Here we apply these ideas to surfaces that are implicitly defined by a smooth function, $Σ=F^{-1}(c)$, by reinterpreting the problem in the context of the metric given by the Hessian of $F$, which is not always positive definite. So, we are naturally led to the study of curves in $E_1^3$. We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the casual character of the curve, present a complete characterization of spherical curves in $E_1^3$, and apply it to characterize curves that belong to a non-degenerate Euclidean quadric. We also interpret the casual character that a curve may assume when we pass from $E^3$ to $E_1^3$ and finally establish a criterion for a curve to lie on a level surface of a smooth function, which reduces to a linear equation when the Hessian is constant.

math.DG↗