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Cesar R. de Oliveira

Publications and source records attributed to Cesar R. de Oliveira.

At least 19 recordsLinked to original sources
arXiv2026

Remarks on Regular Approximations to the Robin Aharonov-Bohm Hamiltonian

In the space $\mathbb R^3$, for Robin parameter $L>0$, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the $Γ$-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.

math-ph↗
arXiv2020

Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model

A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac cones are then proven to be present only for trilayer graphene, although the bilayer has a gapless parabolic band component. Our model rigorously exhibits basic facts from tight-binding calculations, effective two-dimensional models and a $π$-orbital continuum model with nearest-neighbour tunneling that have been discussed in the physics literature.

math-ph↗
arXiv2020

All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schrödinger operator, in a quasi-convex domain~$Ω$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overlineΩ)$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.

math-ph↗
arXiv2020

A new version of the Aharonov-Bohm effect

We propose a simple situation in which the magnetic Aharonov-Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov-Bohm effect.

math-ph↗
arXiv2017

Results on the Aharonov-Bohm effect without contact with the solenoid

We add a confining potential to the Aharonov-Bohm model resulting in no contact of the particle with the solenoid (border); this is characterized by a unique self-adjoint extension of the initial Hamiltonian operator. It is shown that the spectrum of such extension is discrete and the first eigenvalue is found to be a nonconstant 1-periodic function of the magnetic flux circulation with a minimum at integers and maximum at half-integer circulations. This is a rigorous verification of the effect.

math-ph↗
arXiv2012

Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

We restrict a quantum particle under a coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three dimensional tubes along the x-axis and diameter $\varepsilon$, and study the confining limit $\varepsilon\to0$. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.

math-ph↗
arXiv2011

On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.

math-ph↗
arXiv2010

Quantum singular operator limits of thin Dirichlet tubes via $Γ$-convergence

The $Γ$-convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e., the vanishing of the cross section diameter) of the Laplace operator with Dirichlet boundary conditions; a procedure to obtain the effective Schrödinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.

math-ph↗
arXiv2010

Scattering and self-adjoint extensions of the Aharonov-Bohm hamiltonian

We consider the hamiltonian operator associated with planar sec- tions of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior. First, in the Sobolev space $\mathcal H^2$, we characterize all generalized boundary conditions on the solenoid bor- der compatible with quantum mechanics, i.e., the boundary conditions so that the corresponding hamiltonian operators are self-adjoint. Then we study and compare the scattering of the most usual boundary con- ditions, that is, Dirichlet, Neumann and Robin.

math-ph↗
arXiv2009

Quantum Energy Expectation in Periodic Time-Dependent hamiltonians via Green Functions

Let $U_F$ be the Floquet operator of a time periodic hamiltonian $H(t)$. For each positive and discrete observable $A$ (which we call a {\em probe energy}), we derive a formula for the Laplace time average of its expectation value up to time $T$ in terms of its eigenvalues and Green functions at the circle of radius $e^{1/T}$. Some simple applications are provided which support its usefulness.

math-ph↗
arXiv2008

Mathematical justification of the Aharonov-Bohm hamiltonian

It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both $\R^2$ and $\R^3$ spaces.

math-ph↗
arXiv2008

Self-adjoint extensions of Coulomb systems in 1,2 and 3 dimensions

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in $R^n$, n = 1, 2, 3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.

math-ph↗
arXiv2007

Almost Periodic Orbits and Stability for Quantum Time-Dependent Hamiltonians

We study almost periodic orbits of quantum systems and prove that for periodic time-dependent Hamiltonians an orbit is almost periodic if, and only if, it is precompact. In the case of quasiperiodic time-dependence we present an example of a precompact orbit that is not almost periodic. Finally we discuss some simple conditions assuring dynamical stability for nonautonomous quantum system.

math-ph↗
arXiv2007

Dynamical Lower Bounds for 1D Dirac Operators

Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli-Dirac one and, in contrast to the discrete case, critical energies are also found for the continuous Dirac case with positive mass.

math-ph↗
arXiv2007

Hybrid Quasicrystals, Transport and Localization in Products of Minimal Sets

We consider convex combinations of finite-valued almost periodic sequences (mainly substitution sequences) and put them as potentials of one-dimensional tight-binding models. We prove that these sequences are almost periodic. We call such combinations {\em hybrid quasicrystals} and these studies are related to the minimality, under the shift on both coordinates, of the product space of the respective (minimal) hulls. We observe a rich variety of behaviors on the quantum dynamical transport ranging from localization to transport.

math-ph↗
arXiv2007

Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on $\ell^2(\mathbb{Z})$ or $\ell^2(\mathbb{N})$ \[ H^W_{ω,S}=Δ+ λF(S^nω) + W, \quad ω\inΩ, \] where $S$ is a transformation on (compact metric) $Ω$, $F$ a real Lipschitz function and $W$ a (sufficiently fast) power-decaying perturbation. Under certain conditions it is shown that $H^W_{ω,S}$ presents quasi-ballistic dynamics for $ω$ in a dense $G_δ$ set. Applications include potentials generated by rotations of the torus with analytic condition on $F$, doubling map, Axiom A dynamical systems and the Anderson model. If $W$ is a rank one perturbation, examples of $H^W_{ω,S}$ with quasi-ballistic dynamics and point spectrum are also presented.

math-ph↗