Cantor spectrum of graphene in magnetic fields

Abstract

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux Φ/2π through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

On the magnetic Schrödinger hyperbolic equation with randomized coefficients

“…Since last century, electromagnetic theory has been extensively utilized in the graphene research in magnetic fields, medical research of organs' biomagnetism, vortex study in the superconductor which carries quantized magnetic flux, and geographical forecasts for cataclysms, such as earthquakes, volcanic eruptions, geomagnetic reversal, etc. Magnetic Schrödinger type equations are important physical models in these respects [2,4,10,12]. In this background, we discuss the regularity of the solution of the magnetic Schrödinger hyperbolic equation with various types of oscillating coefficients of second-order moment stochastic processes.…”

“…that 𝐄 = −∇𝜙, where the scalar 𝜙 represents the electric potential. Next, we choose an appropriate Lagrangian for the charged particle in the electromagnetic field (𝑞 is the electric charge of the particle, 𝐯 is its velocity and 𝑚 is mass) ℒ = 𝑚𝐯 2 2 − 𝑞𝜙 + 𝑞𝐯 ⋅ 𝐀.…”

“…It is worth noticing that this Hamiltonian operator phenomenologically describes a quantity of behaviors discovered in superconductors and quantum electrodynamics(QED). In these respects, Ginzburg-Landau equations, Schrödinger equations, Dirac equations and the matrix Pauli operator are famous examples concerned with the above Hamiltonian operator [2,18,25,30,31,34].…”

“…Let 𝑈 ⊂ ℝ 𝑁 be a bounded, open and connected Lipschitz domain with the boundary Γ ∈ 𝐶 2 . Assume that the timeindependent vector potential satisfies 𝐀 ∈ (𝐶 1 (Ω)) 𝑁 .…”

“…Assume that the timeindependent vector potential satisfies 𝐀 ∈ (𝐶 1 (Ω)) 𝑁 . In this manuscript, we mainly focus on the magnetic field and simplify the above Hamiltonian into the following magnetic Schrödinger operator ℋ 𝐀 ∶= (𝑖∇ + 𝐀(𝑥)) 2 ∶ 𝐿 2 (𝑈) → 𝐿 2…”