**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Abstract

In this PhD thesis we deal with two mathematical problems arising from quantum mechanics. We consider a spinless non relativistic quantum particle whose configuration space is a two dimensional surface S. We also suppose that the particle feels the effect of an homogeneous magnetic field perpendicular to the surface S. In the first case S = R × SL1, the infinite cylinder of circumference L, corresponding to periodic boundary conditions, while in the second one S = R2. In both cases the particle feels the effect of an additional suitable potential. We are thus left with the study of two specific classes of Schrödinger operators. The operator of the first class generates the dynamics of the particle when it is submitted to an Anderson-type random potential, as well as to a non random potential confining the particle along the cylinder axis in an interval of length L. In this case we describe the spectrum and classify it by the quantum mechanical current carried by the corresponding eigenfunctions. We prove that there are spectral regions in which all the eigenvalues have an order one current with respect to L, and spectral regions where eigenvalues with order one current and eigenvalues with infinitesimal current with respect to L are intermixed. These results are relevant for the theory of the integer quantum Hall effect. The second Schrödinger operator class corresponds to the physical situation where the potential is the sum of a "local" potential and of a potential due to a weak constant electric field F. In this case we show that the resonant states, induced by the electric field, decay exponentially at a rate given by the imaginary part of the eigenvalues of some non self-adjoint operator. Moreover we prove an upper bound on this imaginary part that turns out to be of order exp(-1/F2) as F goes to zero. Therefore the lifetime of the resonant states is at least of order exp(-1/F2).

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related concepts (24)

Quantum mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quan

Cylinder

A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A

Electric potential

The electric potential (also called the electric field potential, potential drop, the electrostatic potential) is defined as the amount of work energy needed per unit of electric charge to move this

Related publications (41)

Loading

Loading

Loading

This thesis is devoted to studying field-theoretical branes in warped geometries, with emphasis on brane excitations and properties of background solutions. Firstly, we examine the features of a model in which our universe is represented by a local string-like defect embedded in a six-dimensional space-time with warped geometry. We demonstrate that in order to satisfy the dominant energy condition, the metric exterior to the defect's core must depend on its thickness. As a result of this dependence, in the limit of the string's thickness going to zero, either the solution no longer localizes the gravity on the defect, or the ratio of the six-dimensional Planck mass to the four-dimensional one diverges. Next, we propose and study a toy model allowing to investigate the phenomenon of quasilocalization. When applied to gravity, our setup can be seen as a (toy) model of a warped geometry in which the graviton is not fully localized on the brane. Studying the tensor sector of metric perturbations around this background, we find that its contribution to the effective gravitational potential is of four-dimensional type 1/r at intermediate scales and that at large scales it becomes 1/rα, 1

A Fast Ion Loss Detector (FILD) was designed, assembled, installed and commissioned for TCV. This is a radially positionable, scintillator based detector that provides information on the 2D fast-ion velocity space lost at the probe's location. The collected particles are collimated inside the probe head and impinge upon a plate coated with a scintillator material that emits light. The photon flux is relayed to two acquisition systems: a sCMOS camera for high spatial resolution measurements of the emission locations in the scintillator, and fast photo-multipliers that allow time-correlation studies of fast-ion losses with high-frequency electromagnetic fluctuations. From its position with respect to the confined plasma and the fast-ion sources on TCV, FILD probes a small portion of the lost fast-ion phase space with high velocity-space and temporal resolution. Therefore, it naturally complements other available diagnostics such as neutral particle analysers, spectroscopy techniques for light emission from energetic particles following CX reactions, or neutron counters, which feature a broader, but less resolved, spatial coverage of fast-ion dynamics.The TCV-FILD design introduces some novelties for exploring new ranges of operation that may be adopted in similar systems for other Tokamaks, such as ITER. Two entrance slits can collect particles that circulate in co- and cntr-plasma current directions. This will be particularly useful when the second NBH system, injecting in the opposite toroidal direction, with particle energies that can excite strong Alfvénic modes, will be operated on TCV. A controlled pneumatic linear actuator radially positions the detector to expose the slits to the particle flux up to 9mm inward of the vessel wall. A plug-in design was conceived to facilitate diagnostic installation. The diagnostic was installed and commissioned during TCV experiments in 2020. The sensitivity of the detector to the local magnetic field line direction was investigated. The direction of the plasma current was found to select which of the two slits may be traversed by lost particles. The operational limits in discharges with NBH with total delivered energies up to 1MJ were assessed with the help of sensors monitoring the temperature of the probe head and cameras detecting visible light emissions resulting from the graphite shield heating by particle fluxes. Using FILD, fast-ion losses were detected for the first time on TCV in plasma discharges exhibiting strong MHD modes. Ejections of energetic ions were found correlated in time with Sawtooth crashes, as a sequence of individual events, and in strong phase coherence, as a continuous loss in time, with magnetic perturbations of a saturated and toroidally rotating magnetic island of a NTM. These observations, in addition to providing initial results on the relevant physics phenomena, stimulating further experiments and comparisons with theory, demonstrate the ability of TCV-FILD to provide valuable information in plasma discharges of interest for fast-ion studies. These measurements and additional information from other TCV diagnostics may now be combined to reconstruct, with tomographic inversion techniques, more of the fast-ion phase space. This will be used to identify the conditions for the excitation/suppression of magnetic instabilities, develop methods for their real-time control with heating and/or shaping actuators and investigate their dependencies on plasma parameters.

,

ESP particle collector (1) for collecting particles in a particle containing gas stream, comprising an inlet section (4), a collector section (6), and an electrode arrangement (8), the inlet section comprising a flow tube (10) defining a gas flow channel (12) therein bounded by a guide wall (24) extending between an entry end (14) and a collector end (16) that serves as an inlet to the collector section (6), the entry end comprising an inlet (28) for the particle gas stream and a sheath flow inlet portion (26) for generating a sheath flow around the particle gas stream, the collector section comprising a housing (18) coupled to the flow tube, and a collector plate (20) mounted therein having a particle collection surface (23). The ESP particle collector is configured to allow optical analysis of the collector plate particle collection surface to measure particles collected thereon. The electrode arrangement comprises at least a base electrode (8a) positioned below the collection surface and a counter-base electrode (8b) positioned at a separation distance L2 above the collection surface such that an electrical field is generated between the electrodes configured to precipitate said particles on the collection surface, wherein the electric field is in a range of 0.1 kV per mm to 1.5 kV per mm, with an absolute voltage on any said electrode that is less than 10 kV, and wherein a ratio ratio I of a radius LI of said inlet at the collector end divided by said separation distance L2 is in a range of 0.8 to 1.2.

2022