Hajo Leschke

CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

The objects of the present study are one-parameter semigroups generated by Schrodinger 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 Schrodinger 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. Simons landmark paper [Bull. Amer. Math. Soc.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

Abstract: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 magnetic field and a random potential which may be unbounded from above and below.In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states.This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schrödinger operators which holds for rather general magnetic fields and different boundary conditions.Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials.Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.

The functional–analytic versus the functional–integral approach to quantum Hamiltonians: The one‐dimensional hydrogen atoma)

The capabilities of the functional–analytic and of the functional–integral approach for the construction of the Hamiltonian as a self‐adjoint operator on Hilbert space are compared in the context of non‐relativistic quantum mechanics. Differences are worked out by taking the one‐dimensional hydrogen atom as an example, that is, a point mass on the Euclidean line subjected to the inverse–distance potential. This particular choice is made with the intent to clarify a long‐lasting discussion about its spectral properties. In fact, for the four‐parameter family of possible Hamiltonians the corresponding energy‐dependent Green functions are derived in closed form. The multiplicity of Hamiltonians should be kept in mind when modeling certain experimental situations as, for instance, in quantum wires.

EXISTENCE AND UNIQUENESS OF THE INTEGRATED DENSITY OF STATES FOR SCHRÖDINGER OPERATORS WITH MAGNETIC FIELDS AND UNBOUNDED RANDOM POTENTIALS

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 Schrodinger 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).

A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids

Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schrodinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.

Changing dimension and time: two well-founded and practical techniques for path integration in quantum physics

In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.

The fate of lifshits 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 nonnegative, algebraically decaying, single-impurity potential we prove that the leading asymptotic behavior 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 eingenspace of any Landau level exhibits the same behavior. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

Operator orderings and functional formulations of quantum and stochastic dynamics

We study the connection between operator ordering schemes and thec-number formulations of quantum mechanics, which are based on generating functionals and functional integrals. We show by explicit construction that different operator ordering schemes are related to different functional and functional integral formulations of quantum mechanics. The results of these considerations are applied to classical non-linear stochastic dynamics by using the formal analogy between the Fokker-Planck equation and the Schrödinger equation.

A Special Case Of A Conjecture By Widom With Implications To Fermionic Entanglement Entropy

We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space

Continuous integral kernels for unbounded Schrödinger semigroups and their spectral projections

L^2(\R^d)