Landau Hamiltonians with Unbounded Random Potentials

We prove the almost sure existence of a pure point spectrum for the two-dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant is assumed to be absolutely continuous. The corresponding densityg has support equal to , and satisfies , for some > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum is , provided the magnetic field B0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum is pure point at all energies except in intervals of width about each lower Landau level , for m < n. We also prove that for any B0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.     

Random laser in one dimension

We present an analytical approach to random lasing in a one-dimensional medium, consistent with transfer matrix numerical simulations. It is demonstrated that the lasing threshold is defined by transmission through the passive medium and thus depends exponentially on the size of the system. Lasing in the most efficient regime of strong three-dimensional localization of light is discussed. We argue that the lasing threshold should have anomalously strong fluctuations from probe to probe, in agreement with recent measurements.     

Phase-space propagators for quantum quadratic Hamiltonians in one and two dimensions

After a summary of the fundamental concepts of quantum mechanics in phase space we apply the Moshinsky-Winternitz classification of the time-independent quadratic Hamiltonians in one and two dimensions to give the explicit form of the phase-space propagators, and make some comments on their spectra.     

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With the spectral parameter, for –<<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for <2 to anomalous behavior for with 2<<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for with <2 to an explicit family with a non-Gaussian covariance for with 2<<4. These non-Gaussian distributions have tails that are broader than Gaussian as varies with 2<<4 and behave for large values like exp(–C _c |x|^4–), withC _c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.     

Random walk in a random medium in one dimension

Applying scaling and universality arguments, the long-time behavior of the probability distribution for a random walk in a one-dimensional random medium satisfying Sinai's constraint is obtained analytically. The convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk.     

Effective Hamiltonians for pseudo—one—dimensional compounds of the type AFeX3

In the past various effective Hamiltonians have been used to describe the properties of pseudo-one-dimensional compounds of the type CsFeX₃ and RbFeX₃ where X is Cl or Br. In the models the number of electronic levels considered have varied from 3 to 25 and some models have included a correlated effective field to incorporate the effects of intrachain exchange interactions. These models are used in fitting the variations of quadrupole splitting with temperature and the results are compared.     

Parallel direct Poisson solver for discretisations with one Fourier diagonalisable direction

In the context of time-accurate numerical simulation of incompressible flows, a Poisson equation needs to be solved at least once per time-step to project the velocity field onto a divergence-free space. Due to the non-local nature of its solution, this elliptic system is one of the most time consuming and difficult to parallelise parts of the code.     

Intertwined Hamiltonians in two-dimensional curved spaces

The problem of intertwined Hamiltonians in two-dimensional curved spaces is investigated. Explicit results are obtained for Euclidean plane, Minkowski plane, Poincaré half plane (AdS₂), de Sitter plane (dS₂), sphere, and torus. It is shown that the intertwining operator is related to the Killing vector fields and the isometry group of corresponding space. It is shown that the intertwined potentials are closely connected to the integral curves of the Killing vector fields. Two problems are considered as applications of the formalism presented in the paper. The first one is the problem of Hamiltonians with equispaced energy levels and the second one is the problem of Hamiltonians whose spectrum is like the spectrum of a free particle.     

Affine Hamiltonians in Higher Order Geometry

Affine Hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of Lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine Hamiltonians and Lagrangians of order k≥2 are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine Hamiltonian h is equivalent with Euler–Lagrange equation of its dual Lagrangian L. Zermelo condition is also studied and some non-trivial examples are given. The authors were partially supported by the CNCSIS grant A No. 81/2005.     

Class of Hamiltonians with one degree-of-freedom allowing application of Kruskal's asymptotic theory in closed form. I

In this paper, apart from a small restriction, all time-dependent Hamiltonians with one degree-of-freedom are determined, for which Kruskal's nice variables can be found by a sort of partial separation of the variables in the equations in question. These Hamiltonians allow an application of Kruskal's perturbation method in closed form, in a way similar to Lewis' treatment of the time-dependent harmonic oscillator. For those “appropriate” Hamiltonians, a connection is further established, between the invariant J following from Kruskal's theory, and an invariant that can be calculated equivalently from Hamilton-Jacobi theory.     

Electrostatic interactions in molecular dynamics simulation of a three-dimensional system with periodicity in one direction

The Mellin transform and Poisson summation formula are used to derive an expression for the Coulomb interaction energy of a three-dimensional system with periodicity in one direction. Initially, calculations are performed for interactions characterized by any inverse power and, using the analytical continuation of the energy function, one obtains the final expression for the interaction energy of charges. We consider also a special case when two different charges are located on a line parallel to the periodicity direction. The energy and force expressions are identical to those obtained from the Lekner summation which is simply a sum over reciprocal lattice terms. The convergence behaviour of the Lekner summation is compared with that based on the Ewald type approach.     

Class of Hamiltonians with one degree-of-freedom allowing application of Kruskal's asymptotic theory in closed form. II

This paper completes a study, started in the companion paper, on the determination of all time-dependent Hamiltonians with one degree-of-freedom for which Kruskal's perturbation method can be applied in closed form, in a way similar to Lewis' treatment of the time-dependent harmonic oscillator.     

Some results in non-commutative ergodic theory

We study some properties of invariant states on aC*-algebraA with a groupG of automorphisms. Using the concept ofG-factorial state, which is a non-commutative generalization of the concept of ergodic measure, in general wider in scope thanG-ergodic state, we show that under a certain abelianity condition on (A,G), which in particular holds for the quasi-local algebras used in statistical mechanics, two differentG-ergodic states are disjoint. We also define the concept ofG-factorial linear functional, and show that under the same abelianity condition such a functional is proportional to aG-ergodic state. This generalizes an earlier result for complex ergodic measures.     

Significance of model Hamiltonians in energy-band theory

The results of recent ab initio calculations of energy bands are compared with interpolation schemes in order to assess the significance of the latter. When the parameters in the interpolation scheme are required to correspond to those obtainable by an ab initio calculation based on a reasonable one-electron potential, we describe the scheme as a model Hamiltonian. The analytic character of model Hamiltonians for s–p bands (pseudopotential case) or for d bands interacting with s–p bands (resonance case) is summarized. The compatibility of the results of several workers with these analyticity requirements is considered. It is shown that if the empirically-motivated adjustments of parameters are to be regarded as physically significant, these adjustments must satisfy rather strict conditions.     

New quasi-exactly solvable Hamiltonians in two dimensions

Quasi-exactly solvable Schrödinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators—the hidden symmetry algebra. In this paper we develop some general techniques for constructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras.Supported in Part by DGICYT Grant PS 89-0011.Supported in Part by an NSERC Grant.Supported in Part by NSF Grant DMS 92-04192.     

A complete solution to the Mössbauer problem, all in one place

We present a full solution to the general combined interactions static Mössbauer problem that is easily generalized to any Mössbauer isotope, and applies for M1, E1, and E2 transitions as well as combined M1–E2 transitions. Explicit expressions are given for both powder and single crystal samples.