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# Magnetic Transport in a Straight Parabolic Channel

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*in*Journal of Physics A General Physics 34(45) · March 2001

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DOI: 10.1088/0305-4470/34/45/312 · Source: arXiv

Cite this publicationAbstract

We study a charged two-dimensional particle confined to a straight parabolic-potential channel and exposed to a homogeneous magnetic field under influence of a potential perturbation $W$. If $W$ is bounded and periodic along the channel, a perturbative argument yields the absolute continuity of the bottom of the spectrum. We show it can have any finite number of open gaps provided the confining potential is sufficiently strong. However, if $W$ depends on the periodic variable only, we prove by Thomas argument that the whole spectrum is absolutely continuous, irrespectively of the size of the perturbation. On the other hand, if $W$ is small and satisfies a weak localization condition in the the longitudinal direction, we prove by Mourre method that a part of the absolutely continuous spectrum persists.

- ... It is well known that the operator P (b, ω) is essentially self-adjoint on C ∞ 0 (R 2 ) (see [14, 21, 22]). For V ≡ 0, it was shown that the spectrum of the unperturbed operator P 0 (b, ω) is absolutely continuous and equal to the interval [0, ∞) (see [22]). ...... Recently a substantial progress has been made in the analysis of the magnetic Schrödinger operators with long-range perturbations going to 0 as |x| → +∞ and the works around the trace formulae have generated many results on the distribution of eigenvalues near Landau levels and Weyl's formula with sharp remainder estimate of the counting function of eigenvalues (see [1, 5, 8, 9, 11, 14, 16, 18, 19, 23, 24, 25, 26, 29] and the references given there). To our best knowledge, there are only a few works concerning the model (1.1) (see [14, 21, 22]). ...... Recently a substantial progress has been made in the analysis of the magnetic Schrödinger operators with long-range perturbations going to 0 as |x| → +∞ and the works around the trace formulae have generated many results on the distribution of eigenvalues near Landau levels and Weyl's formula with sharp remainder estimate of the counting function of eigenvalues (see [1, 5, 8, 9, 11, 14, 16, 18, 19, 23, 24, 25, 26, 29] and the references given there). To our best knowledge, there are only a few works concerning the model (1.1) (see [14, 21, 22]). In [22], the authors studied a quadratic Hamiltonian without perturbation by using the theory of metapletic representations. ...ArticleFull-text available
- Jan 2012

- ... The study of the quantum motion of a charged particle in a two-dimensional medium submitted to an orthogonal magnetic field of constant strength is at the center of theoretical explanation of edge currents in Hall systems, and is a source of interesting spectral problems. Some of them have been rigourously investigated by many authors in recent years ([13,6,20,12,8,4,9,10,11,2,15,16]). Edge currents have some connection with the integer quantum Hall effect. ...... This need not be the case, however, for more complicated edge geometries. For those situations, there may be edge currents but the spectrum need not be absolutely continuous (cf.[8,9,10,11,4,16]). For instance, in the case of a strip of finite width, adding a second edge radically changes the picture observed for one-edge geometries, since the Hall current has different signs on opposite edges. ...... One of the benefits of a local positive commutator of this type is its stability under perturbation. It is therefore particularly useful to prove the persistence of edge currents in presence of weak disorder (cf.[8,4,16]). ...ArticleFull-text available
- Dec 2010

We consider a 2D Schrödinger operator H_0 with constant magnetic field defined on a strip of finite width. The spectrum of H_0 is absolutely continuous and contains a discrete set of thresholds. We perturb H_0 by an electric potential V, and establish a Mourre estimate for H = H_0 + V when V is periodic in the infinite direction of the strip, or decays in a suitable sense at infinity. In the periodic case, for each compact subinterval I contained in between two consecutive thresholds, we show as a corollary that the spectrum of H remains absolutely continuous in I, provided the period and the size of the perturbation are sufficiently small. In the second case we obtain that the singular continuous spectrum of H is empty, and any compact subset of the complement of the thresholds set contains at most a finite number of eigenvalues of H, each of them having finite multiplicity. Moreover these Mourre estimates together with some of their spectral consequences generalize to the case of 2D magnetic Schrödinger operators defined on the plane for suitable confining potentials modeling Dirichlet boundary conditions. - ... As a preamble to the investigation of these models, we shall systematically examine the straight parabolic channel model studied by Exner, Joye and Kovarik in [2]. In this case the confining potential is defined by ...... As a warm up, we address now the model studied by Exner, Joye and Kovarik in [2], where the confining potential is given by (1.4). For this model, the electron is confined to a parabolic channel of infinite extent in the y-direction. ...... We prove that the spectrum of H is purely absolutely continuous if 1) V 1 (x, y) is periodic with respect to y with sufficiently small period or 2) V 1 (x, y) has some decay in y-direction. These results are similar to those of Exner, Joye, and Kovarik [2]. We point out that for the more general class of perturbations V 1 treated in sections 4 and 5 of [8], such as random potentials, we do not know the spectral type of the operator H. ...Article
- Jun 2008
- ANN HENRI POINCARE

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator. - ... It is well-known that the operator P (B, ω) is essentially self-adjoint on C ∞ 0 (R 2 ), (see [12] [23]). For V ≡ 0, ω = 0, it was shown that the spectrum of the unperturbed Hamiltonian P 0 (B, 0) := (D x − By) 2 + D 2 y consists of eigenvalues λ n = (2n + 1)B, n ∈ N with infinite multiplicities called Landau levels (see [1] [7] [24] [25]). ...... By applying the Weyl theorem (see [15]) the essential spectrum of P (B, ω) is equal to that of P 0 (B, ω). Further, the absolutely continuous spectrum of P (B, ω) was investigated in [12]. Recently, the counting function of discrete eigenvalues of P (B, ω) in (−∞, √ B 2 + ω 2 ) has been studied in [10]. ...ArticleFull-text available
- Jan 2012

The purpose of this paper is to study the Schr\"odinger operator $P(B,\omega)=(D_x-By)^2+D_y^2+\omega^2x^2+V(x,y),\;(x,y)\in{\mathbb R}^2,$ with the magnetic field $B$ large enough and the constant $\omega\not=0$ is fixed and proportional to the strength of the electric field. Under certain assumptions on the potential $V$, we prove the existence of resonances near Landau levels as $B\to\infty$. Moreover, we show that the width of resonances is of size $\mathcal{O}\Big(B^{-\infty}\Big)$. - ... The one-electron model approximation, although certainly insufficient to explain all the aspects of Hall quantization, in particular the fractional quantum Hall effect, is nevertheless a source of interesting spectral problems. Some of them have been rigorously investigated by various authors ( [11], [12], [13], [14], [15]). In this paper we will summarize and complement existing results about existence and properties of current carrying edge states and point out some open problems. ...... Finally, for the strip geometry, L 0 < +∞, there are very few results. Recently it was shown in [14] that absolutely continuous spectrum of H 0 survives perturbation by V if V is periodic or decays fast enough in y-direction. ...Article
- May 2002
- COMMUN CONTEMP MATH

The study of the quantum motion of a charged particle in a half-plane as well as in an infinite strip submitted to a perpendicular constant magnetic field reveals eigenstates propagating permanently along the edge, the so-called edge states. Moreover, in the half-plane geometry, current carried by edge states with energy in between the Landau levels persists in the presence of a perturbating potential small relative to the strengh of the magnetic field. We show here that edge states carrying current survive in an infinite strip for a long time before tunneling between the two edges has a destructive effect on it. The proof relies on Helffer-Sjöstrand functional calculus and decay properties of quantum Hall Hamiltonian resolvent. - ... Whereas Iwatsuka needed some estimates on the growth of the eigenfunctions to show that the asymptotic behaviour of the eigenvalues in ±∞ is determined by that of the magnetic field, we derive the asymptotic behaviour of the eigenvalues directly using some comparison argument based on the minimax principle combined with a norm-resolvent convergence result. Iwatsuka's model with a non-zero electric field (W = 0) of a particular type has been studied before in [5]. There it was proved that H L + ω 2 x 2 remains purely absolutely continuous under a perturbation that is either a bounded function of x–variable only or a bounded periodic function of y–variable only. ...Article
- Sep 2016
- J PHYS A-MATH THEOR

We prove absolute continuity for an extended class of two-dimensional magnetic Hamiltonians that was initially studied by A. Iwatsuka. In particular, we add an electric field that is translation invariant in the same direction as the magnetic field is. The proof relies on an abstract convergence result that may be of independent interest. As an example we study the effective Hamiltonian for a thin quantum layer in a homogeneous magnetic field. - ... The spectrum of periodically and randomly curved waveguides was investigated in [76, 74] and [55], respectively. Finally, let us mention systems where Ω = R n , n = 2, 3, and the quantum waveguide is introduced by means of a magnetic field [28, 36, 29] or a strong Dirac interaction supported by an infinite curve or surface [27, 48, 30, 31, 24, 32, 33, 34, 35, 41] . The present paper is devoted to a study of the interplay between the geometry , boundary conditions (we consider uniform Dirichlet, Neumann or a combination of these ones) and the spectral properties of the Laplacian in the infinite planar curved strips. ...Article
- Sep 2005
- PUBL RES I MATH SCI

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides. - ... For those situations, there may be edge currents carried by states ψ but the spectrum need not be absolutely continuous (cf. [27, 16, 18, 19, 20]). ...Article
- Feb 2008
- REV MATH PHYS

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions. - Article
- May 2017
- J GEOM ANAL

We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/$\epsilon$. Under a gap condition on the fibre-wise eigenvalues we prove existence of effective operators that provide asymptotics to any order in $\epsilon$ for H (with Dirichlet boundary conditions), on an appropriate almost-invariant subspace of L${}^2$(E). - ArticleFull-text available
- Jan 2004

- ChapterFull-text available
- Sep 2006

We review recent results of two of the authors concerning the quantization of Hall currents, in particular a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. We then study the case where the Fermi energy is located in a region of localized states and discuss new regularizations. We also sketch the proof of localization for 2D-models with constant magnetic field with random potential located in a halfplane in two different situations: (1) with a zero potential in the other half plane and for energies away from the Landau levels and (2) with a confining potential in the other half plane and on an interval of energies that covers an arbitrary number of Landau levels. - ArticleFull-text available
- Apr 2002

In this note we review spectral properties of magnetic random Schrodinger operators H ! = H 0 + V ! + U ` + U r de ned on L ; dxdy) with periodic boundary conditions along y. U ` and U r are two con ning potentials for x 2 and x respectively and vanish for x . We describe the spectrum in two energy intervals and we classify it according to the quantum mechanical current of eigenstates along the periodic direction. The rst interval lies in the rst Landau band of the bulk Hamiltonian, and contains intermixed eigenvalues with a quantum mechanical current of O(1) and O respectively. The second interval lies in the rst spectral gap of the bulk Hamiltonian, and contains only eigenvalues with a quantum mechanical current of O(1). - Article
- Aug 2003
- LETT MATH PHYS

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schr\"odinger operators with a sufficiently strongly attractive $\delta$ interaction supported by a periodic curve in $\mathbb{R}^d, d=2,3$. - Article
- Apr 2003
- Waves Random Media

We discuss of a ring-shaped soft quantum wire modeled by $\delta$ interaction supported by the ring of a generally nonconstant coupling strength. We derive condition which determines the discrete spectrum of such systems, and analyze the dependence of eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is investigated also in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov-Bohm flux and the system exhibits persistent currents. - We study edge states of a random Schroedinger operator for an electron submitted to a magnetic field in a finite macroscopic two dimensional system of linear dimensions equal to L. The y direction is L-periodic and in the x direction the electron is confined by two smoothly increasing parallel boundary potentials. We prove that, with large probability, for an energy range in the first spectral gap of the bulk Hamiltonian, the spectrum of the full Hamiltonian consists only on two sets of eigenenergies whose eigenfuntions have average velocities which are strictly positive/negative, uniformly with respect to the size of the system. Our result gives a well defined meaning to the notion of edge states for a finite cylinder with two boundaries, and extends previous studies on systems with only one boundary.

- We review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved. We point out a relationship between the virial theorem for a pair of operators H, A and the regularity properties of the map . We give an example showing that the statement of the virial theorem in [CFKS] is incorrect.
- Jan 1994

- J Blank
- P Exner
- M Havlíček

Blank J, Exner P and Havlíček M 1994 Hilbert Space Operators in Quantum Physics (New York: AIP) [BP] deBì evre S and Pulé J V 1999 Propagating edge states for a magnetic Hamiltonian Math. Phys. Electron. J. 5- Jan 1999
- 124-154

- P Exner
- Joye A Kovařík

Exner P, Joye A and Kovařík H 1999 Edge currents in the absence of edges Phys. Lett. A 264 124–30 [EK]- Hall conductance, current carrying edge states, and the existence of extended states in two-dimensional disordered potential Phys. Rev. B 25 2185–90 [Iw] Iwatsuka A 1985 Examples of absolutely continuous Schrödinger operators in magnetic fields Publ
- Jan 1982
- 385-401

- B Halperin

Halperin B I 1982 Quantized Hall conductance, current carrying edge states, and the existence of extended states in two-dimensional disordered potential Phys. Rev. B 25 2185–90 [Iw] Iwatsuka A 1985 Examples of absolutely continuous Schrödinger operators in magnetic fields Publ. RIMS 21 385–401 [Ka] - Jan 1975

- M Reed

Reed M and Simon B 1975 Methods of Modern Mathematical Physics: II. Fourier Analysis, Self-Adjointness (New York: Academic)- Jan 1966

- T Kato

Kato T 1966 Perturbation Theory for Linear Operators (Heidelberg: Springer) [Mo]- Jan 2000
- 3297-311

- P Exner
- H Kovařík

Exner P and Kovařík H 2000 Magnetic strip waveguides J. Phys. A: Math. Gen. 33 3297–311- Article
- Jan 1997
- J OPERAT THEOR

We develop a version of the conjugate operator method for an arbitrary pair of selfadjoint operators: the Hamiltonian H and the conjugate operator A. We obtain optimal results concerning the regularity properties of the boundary values (H-λ∓i0) -1 of the resolvent of H as functions of λ. Our approach allows one to eliminate the spectral gap hypothesis on H without asking the invariance of the domain or of the form domain of the Hamiltonian under the unitary group generated by A (previous versions of the theory assume at least one of theses conditions). In particular, one may treat singular Hamiltonians with spectrum equal ℝ, e.g. strong singular perturbations of Stark Hamiltonians or simply characteristic operators. - Book
- Jan 1996

Preface.- Comments on notations.- 1 Some Spaces of Functions and Distributions.- 2 Real Interpolation of Banach Spaces.- 3 C0-Groups and Functional Calculi.- 4 Some Examples of C0-Groups.- 5 Automorphisms Associated to C0-Representations.- 6 Unitary Representations and Regularity.- 7 The Conjugate Operator Method.- 8 An Algebraic Framework for the Many-Body Problem.- 9 Spectral Theory of N-Body Hamiltonians.- 10 Quantum-Mechanical N-Body Systems.- Bibliography.- Notations.- Index. - Article
- Feb 1982
- Phys Rev B

When a conducting layer is placed in a strong perpendicular magnetic field, there exist current-carrying electron states which are localized within approximately a cyclotron radius of the sample boundary but are extended around the perimeter of the sample. It is shown that these quasi-one-dimensional states remain extended and carry a current even in the presence of a moderate amount of disorder. The role of the edge states in the quantized Hall conductance is discussed in the context of the general explanation of Laughlin. An extension of Laughlin's analysis is also used to investigate the existence of extended states in a weakly disordered two-dimensional system, when a strong magnetic field is present. - Article
- Feb 1984
- PHYS REV B

It is shown that the quantized Hall current may always be expressed as the difference between diamagnetic currents flowing at the two edges. It is argued that the high precision of the quantization may be aided by the establishment of a local equilibrium in each edge region. The basic ideas are illustrated by the discussion of a free two-dimensional electron gas in an infinite confining potential. Our derivation establishes the connection between quantum-mechanical and classical thermodynamic explanations for the quantum Hall effect. - Article
- Sep 1999
- PHYS REV B

The boundary element method for electron waves in the presence of uniform magnetic fields is extended so that it is applicable to scattering problems by many scatterers. The cross sections of scatterers are assumed to be so small that the scattering potentials are modeled by summation of δ functions. This extended method is applied to a magnetic electron focusing geometry with a sequence of scatterers between an emitter and a collector. The transmission probability of an electron wave from the emitter to the collector is computed as a function of the magnetic field. Electron distributions are also calculated. These evidently show the commensurate scattering classically expected. The results of sample calculations demonstrate the effectiveness of the method. - Jan 1978

- M Reed

Reed M and Simon B 1978 Methods of Modern Mathematical Physics: IV. Analysis of Operators (New York: Academic) [Sa1]- Article
- Jul 1981
- COMMUN MATH PHYS

We give a sufficient condition for a self-adjoint operator to have the following properties in a neighborhood of a pointE of its spectrum:a) its point spectrum is finite; b) its singular continuous spectrum is empty; c) its resolvent satisfies a class of a priori estimates. - Article
- Oct 1997
- COMMUN MATH PHYS

In this paper we study a two dimensional magnetic field Schrödinger Hamiltonian introduced in [7]. This model has some interesting propagation properties, as conjectured in [2] and at the same time is a special case of the class of analytically decomposable Hamiltonians [5]. Our aim is to start from a conjugate operator, intimately related to the band structure of the Hamiltonian and to prove existence of an asymptotic velocity in one spatial direction and a theorem giving minimal and maximal velocity bounds for the propagation associated to the Hamiltonian. A simple example of this model, with a very simple conjugate operator, has been given in [9]. At the same time, by using the Virial Theorem, we obtain a generalisation of the hypothesis in [7]. - Article
- Oct 1998
- ANN PHYS-NEW YORK

We construct raising and lowering operators for certain orthonormal bases ofL2(n). These bases consist of quantum mechanical wave packets that can be used to develop asymptotic expansions for solutions to the time–dependent Schrödinger equation in the semiclassical limit. With the knowledge of the raising and lowering operators, we simplify the construction of these bases and the proofs of their crucial properties. We also present simplified proofs of several results in semiclassical quantum mechanics. - Article
- Jan 1985
- PUBL RES I MATH SCI

On considere l'operateur de Schrodinger a 2 dimensions H qui est la realisation auto-adjointe dans #7B-H=L 2 (R 2 ) de l'operateur differentiel L=(∂/i∂x−a) 2 +(∂/i∂y−b) 2 ou a et b sont les operateurs de multiplication par des fonctions C ∞ a valeur reelle a (x,y) et b(x,y) - Article
- Dec 1973
- COMMUN MATH PHYS

A time dependent scattering theory for a quantum mechanical particle moving in an infinite, three dimensional crystal with impurity is given. It is shown that the Hamiltonian for the particle in the crystal without impurity has only absolutely continuous spectrum. The domain of the resulting wave operators is therefore the entire Hilbert space. - Article
- Jul 2000
- J MATH PHYS

We study the spectral properties of the Schrodinger operator with a constant electric field perturbed by a bounded potential. It is shown that if the derivative of the potential in the direction of the electric field is smaller at infinity than the electric field then the spectrum of the corresponding Stark operator operator is purely absolutely continuous. In one dimension the absolute continuity of the spectrum is implied by just the boundedness of the derivative of the potential. The sharpness of our criterion for higher dimensions is illustrated by constructing smooth potentials with bounded partial derivatives for which the corresponding Stark operators have a dense point spectrum. This work was partially supported by the NSF Grant PHY-9971149. Permanent Adress: Universit'e Paris 7, U.F.R. de Math'ematiques, 2, place Jussieu, 75251 Paris Cedex 05, e-mail : sahbani@math.jussieu.fr 1 I Introduction I.1. Overview The Hamiltonian of an electron moving in a constant electric fiel... - Article
- Apr 1999

We study the quantum mechanical motion of a charged particle moving in a half plane (x>0) subject to a uniform constant magnetic field B directed along the z-axis and to an arbitrary impurity potential W_B, assumed to be weak in the sense that ||W_B||_\infty < \delta B, for some \delta small enough. We show rigorously a phenomenon pointed out by Halperin in his work on the quantum Hall effect, namely the existence of current carrying and extended edge states in such a situation. More precisely, we show that there exist states propagating with a speed of size B^{1/2} in the y-direction, no matter how fast W_B fluctuates. As a result of this, we obtain that the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size \gamma B (for some \gamma <1) between the Landau levels of the unperturbed system (i.e. the system without edge or potential), so that the corresponding eigenstates are extended. - Article
- Apr 1999
- ANN HENRI POINCARE

Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, the Mourre theory of positive commutators is applied to prove absolute continuity of the energy spectrum well in between Landau levels, corresponding to edge states. - Article
- Nov 1999
- J Phys Math Gen

We analyze the spectrum of the "local" Iwatsuka model, i.e. a two-dimensional charged particle interacting with a magnetic field which is homogeneous outside a finite strip and translationally invariant along it. We derive two new sufficient conditions for absolute continuity of the spectrum. We also show that in most cases the number of open spectral gaps of the model is finite. To illustrate these results we investigate numerically the situation when the field is zero in the strip being screened, e.g. by a superconducting mask. - We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current. The fact that the presence of boundaries can induce a transport in a system with a homogeneous magnetic field is known for long [1, 2] and has numerous consequences in solid state physics. The mentioned pioneering papers were followed by tremendous number of studies in which the magnetic transport was analyzed numerically in various models as well as experimentally. The obstacles at which the particle “bounces ” need not be hard walls but also objects with openings such as various antidot lattices; for a sample of literature
- We consider an electron in two dimensions submitted to a magnetic field and to the potential of impurities. We show that when the electron is confined to a half-space by a planar wall described by a smooth increasing potential, the total Hamiltonian necessarily has a continuous spectrum in some intervals in-between the Landau levels provided that both the amplitude and spatial variation of the impurity potential are sufficiently weak. The spatial decay of the impurity potential is not needed. In particular this proves the occurence of edge states in semi-infinite quantum Hall systems. Comment: 20 pages, no figures, plain TEX, to appear in J. Phys. A