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Hajo Leschke

Publications and source records attributed to Hajo Leschke.

31 records · Page 2Linked to original sources
arXiv2000

Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

A detailed mathematical proof is given that the energy spectrum of a non-relativistic quantum particle in multi-dimensional Euclidean space under the influence of suitable random potentials has almost surely a pure-point component. The result applies in particular to a certain class of zero-mean Gaussian random potentials, which are homogeneous with respect to Euclidean translations. More precisely, for these Gaussian random potentials the spectrum is almost surely only pure point at sufficiently negative energies or, at negative energies, for sufficiently weak disorder. The proof is based on a fixed-energy multi-scale analysis which allows for different random potentials on different length scales.

math-ph↗
arXiv2000

Is the Universe a Quantum System?

In order to relate the probabilistic predictions of quantum theory uniquely to measurement results, one has to conceive of an ensemble of identically prepared copies of the quantum system under study. Since the universe is the total domain of physical experience, it cannot be copied, not even in a thought experiment. Therefore, a quantum state of the whole universe can never be made accessible to empirical test. Hence the existence of such a state is only a metaphysical idea. Despite prominent claims to the contrary, recent developments in the quantum-interpretation debate do not invalidate this conclusion.

quant-ph↗
arXiv2000

Continuity properties of Schrödinger semigroups with magnetic fields

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc. (N.S.) {\bf 7}, 447--526 (1982)] to non-zero vector potentials and more general configuration spaces.

math-ph↗
arXiv1999

The multiformity of Lifshits tails caused by random Landau Hamiltonians with repulsive impurity potentials of different decay at infinity

For a charged quantum particle in the Euclidean plane subject to a perpendicular constant magnetic field and repulsive impurities, randomly distributed according to Poisson's law, we determine the leading low-energy fall-off of the integrated density of states in case the single-impurity potential has either super-Gaussian or regular sub-Gaussian long-distance decay. The forms of the resulting so-called magnetic Lifshits tails reflect the great variety of these decays. On the whole, we summarize, unify, and generalize results in previous works of K. Broderix, L. Erdős, D.\ Hundertmark, W. Kirsch, and ourselves.

math-ph↗
arXiv1999

A rigorous path integral for quantum spin using flat-space Wiener regularization

Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on $R^2$. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.

math-ph↗
arXiv1998

A rigorous path-integral formula for quantum-spin dynamics via planar Brownian motion

Adapting ideas of Daubechies and Klauder we derive a continuum path-integral formula for the time evolution generated by a spin Hamiltonian. For this purpose we identify the finite-dimensional spin Hilbert space with the ground-state eigenspace of a suitable Schödinger operator on $L^2({\mathbb{R}}^2)$, the Hilbert space of square-integrable functions on the Euclidean plane ${\mathbb{R}}^2$, and employ the Feynman-Kac-Itô formula.

quant-ph↗
arXiv1998

On the averaged quantum dynamics by white-noise Hamiltonians with and without dissipation

Exact results are derived on the averaged dynamics of a class of random quantum-dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time-independent and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white-noise system is shown to be a monotone mixing increasing quantum-dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white-noise system linearly to a quantum-mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.

quant-ph↗
arXiv1995

Classical and quantum dynamics of the n-dimensional kicked rotator

The classical and quantum dynamics for an n-dimensional generalization of the kicked planar (n=1) rotator in an additional effective centrifugal potential. Therefore, typical phenomena like the diffusion in classical phase space are similar to that of the one-dimensional model. For the quantum dynamics such a result is not expected as in this case the evolution does depend in a very complicated way on the number n of degrees of freedom. In the limit n --> infinity we find the free undistrubed quantum motion. For finite values of n (1<=n<=26) we study numerically the quantum dynamics. Here, we always find localization independent of the actual number of degrees of freedom.

chao-dyn↗
arXiv1995

Lowest Landau level broadened by a Gaussian random potential with an arbitrary correlation length: An efficient continued-fraction approach

For an electron in the plane subjected to a perpendicular constant magnetic field and a homogeneous Gaussian random potential with a Gau{ss}ian covariance function we approximate the averaged density of states restricted to the lowest Landau level. To this end, we extrapolate the first 9 coefficients of the underlying continued fraction consistently with the coefficients' high-order asymptotics. We thus achieve the first reliable extension of Wegner's exact result [Z. Phys. B {\bf 51}, 279 (1983)] for the delta-correlated case to the physically more relevant case of a non-zero correlation length.

cond-mat↗
arXiv1995

The Fate of Lifshitz Tails in Magnetic Fields

We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

cond-mat↗
arXiv1994

Dynamics by White-Noise Hamiltonians

A new class of random quantum--dynamical systems in continuous space is introduced and studied in some detail. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time--independent, and quadratic, the Weyl--Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both(mixed) states and observables averaged over randomness are obtained.The differences between the quantum and the classical behavior are clearly exhibited.As a special case it is shown that, if the deterministic part corresponds to a particle subjected to a constant magnetic field,the spatial variance of the averaged state grows diffusively for long times independent of the initial state.

cond-mat↗