64 research outputs found
Simone WARZEL - Mathematical challenges and results related to fractional quantum systems: AMS-EMS-SMF International meeting 2022
No full textIn this talk, I will give an overview over recent results and open problems related to key properties of fractional Hall systems. The ground state properties of such systems are famously described by Laughlin wavefunctions. Curiously, the latter are related to a 2D classical electrostatic problem known as jellium. Although this classical system already features one key property of fractional Hall systems, namely their incompressibility, a microscopic explanation of this phenomenon requires to establish a spectral gap in associated many-particle operators such as the Haldane pseudopotentials, which I will discuss. In particular, I will survey the recent mathematical results and methods concerning Haldane pseudopotentials in a thin cylinder geometry
EXISTENCE AND UNIQUENESS OF THE INTEGRATED DENSITY OF STATES FOR SCHRÖDINGER OPERATORS WITH MAGNETIC FIELDS AND UNBOUNDED RANDOM POTENTIALS
No full text The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volume limits of spatial eigenvalue concentrations of finite-volume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinite-volume operator, the integrated density of states is almost surely non-random and independent of the chosen boundary condition. Our proof of the independence of the boundary condition builds on and generalizes certain results obtained by S. Doi, A. Iwatsuka and T. Mine (Math. Z. 237 (2001) 335) and S. Nakamura (J. Funct. Anal. 173 (2001) 136). </jats:p
Resonant delocalization in Random Schrödinger operators
No full textNon UBCUnreviewedAuthor affiliation: TU MunichFacult
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Transport in Multi-Dimensional Random Schrödinger Operators
No full textRandom Schrödinger operators are a topic of common interest i
Hierarchische Zufallsmatrizen und Operatoren
No full textWe consider three hierarchical random matrices and operators. First, we implement a renormalization group for the hierarchical Anderson model to prove dynamical localization and Poisson statistics in all spectral dimensions. Next, we study the local stability of Dyson Brownian motion to map out the entire localized phase of the ultrametric ensemble in terms of both eigenfunctions and local statistics. Finally, we study the characteristics of a certain stochastic advection equation to prove the existence of a hitherto conjectured non-ergodic phase in the Rosenzweig-Porter model.Wir betrachten drei hierarchische Zufallsmatrizen und Operatoren. Zuerst implementieren wir eine Renormierungsgruppe für das hierarchische Anderson-Modell, um dynamische Lokalisierung und Poisson-Statistik in allen spektralen Dimensionen zu beweisen. Dann untersuchen wir die lokale Stabilität der Dyson-Brown’schen Bewegung, um das gesamte Lokalisierungsregime des ultrametrischen Ensembles abzubilden. Schließlich betrachten wir die charakteristischen Kurven einer bestimmten stochastischen Transportgleichung, um die Existenz einer bisher vermuteten nicht-ergodischen Phase im Rosenzweig-Porter-Modell zu beweisen
Transport in Multi-Dimensional Random Schrödinger Operators
No full textRandom Schrödinger operators are a topic of common interest i
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Correlations and Interactions for Random Quantum Systems
No full textRandom quantum systems cover a broad range of mathematical models from random Schr¨odinger operators to random matrices and quantum spin models with random parameters. Their understanding requires techniques which combine functional analysis and probability. The workshop brought together researchers from these various branches which discussed new results, methods and future challenges. This is a report on the meeting containing extended abstracts of the lectures
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Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization
No full textDiscussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K-property, the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrodinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon-Wolff rank-one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.(c) 2015 Wiley Periodicals, Inc
Anderson localization and Lifshits tails for random surface potentials
No full textABSTRACT. We consider Schrödinger operators on L 2 (R d) with a random potential concentrated near the surface R d1 × {0} ⊂ R d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail rlies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band o
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Kac-Ward Formula and Its Extension to Order-Disorder Correlators Through a Graph Zeta Function
No full textA streamlined derivation of the Kac-Ward formula for the planar Ising model’s partition function is presented and applied in relating the kernel of the Kac-Ward matrices’ inverse with the correlation functions of the Ising model’s order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on Z(2) for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature
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