arXiv subjects

Daniel M. Elton

Publications and source records attributed to Daniel M. Elton.

8 recordsLinked to original sources

arXiv2017

Decay rates at infinity for solutions to periodic Schrödinger equations

We consider the equation $Δu=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation $Δu=Vu$ is studied as part of a broader class of elliptic evolution equations.

math.AP↗

arXiv2017

On the non-existence of zero modes

We consider magnetic fields on $\mathbb{R}^3$ which are parallel to a conformal Killing field. When the latter generates a simple rotation we show that a Weyl-Dirac operator with such a magnetic field cannot have a zero mode. In particular this allows us to expand the class of non zero mode producing magnetic fields to include examples of non-trivial smooth compactly supported fields.

math.SP↗

arXiv2015

Asymptotics for Erdos-Solovej Zero Modes in Strong Fields

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl-Dirac operators on $\mathbb{R}^3$. In particular we are interested in those operators $\mathcal{D}_{B}$ for which the associated magnetic field $B$ is given by pulling back a $2$-form $β$ from the sphere $\mathbb{S}^2$ to $\mathbb{R}^3$ using a combination of the Hopf fibration and inverse stereographic projection. If $\int_{\mathbb{S}^2}β\neq0$ we show that \[ \sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB} =\frac{T^2}{8π^2}\,\biggl\lvert\int_{\mathbb{S}^2}β\biggr\rvert\,\int_{\mathbb{S}^2}\lvertβ\rvert+o(T^2) \] as $T\to+\infty$. The result relies on Erdős and Solovej's characterisation of the spectrum of $\mathcal{D}_{tB}$ in terms of a family of Dirac operators on $\mathbb{S}^2$, together with information about the strong field localisation of the Aharonov-Casher zero modes of the latter.

math-ph↗

arXiv2015

Approximate Zero Modes for the Pauli Operator on a Region

Let $\mathcal{P}_{Ω,tA}$ denoted the Pauli operator on a bounded open region $Ω\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=\mathrm{curl}\,A$ satisfies $B\in L\log L(Ω)\cap C^α(Ω_0)$ where $α>0$ and $Ω_0$ is an open subset of $Ω$ of full measure (note that, the Orlicz space $L\log L(Ω)$ contains $L^p(Ω)$ for any $p>1$). Let $\mathsf{N}_{Ω,tA}(λ)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula \[ \mathsf{N}_{Ω,tA}(λ(t))=\frac{t}{2π}\int_Ω\lvert B(x)\rvert\,dx\;+o(t) \] as $t\to+\infty$, whenever $λ(t)=Ce^{-ct^σ}$ for some $σ\in(0,1)$ and $c,C>0$. The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.

math.SP↗

arXiv2013

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

math.SP↗

arXiv2003

Asymptotics for the Eigenvalues of the Harmonic Oscillator with a Quasi-Periodic Perturbation

We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form λ_n(H+V)=λ_n(H)+W(\sqrt n)n^{-1/4}+O(n^{-1/2}\ln(n)) as n\to+\infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.

math.SP↗

arXiv2000

Fredholm Properties of Elliptic Operators on $\R^n$

We study the Fredholm properties of a general class of elliptic differential operators on $\R^n$. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard function spaces, and a related spectral pencil problem on the sphere, which is defined in terms of the asymptotic behaviour of the coefficients of the original operator.

math.AP↗

arXiv1998

The Dirac equation without spinors

In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.

math-ph↗