Raymond Brummelhuis

Effective Hamiltonians for atoms in very strong magnetic fields

We propose three effective Hamiltonians which approximate atoms in very strong homogeneous magnetic fields B modelled by the Pauli Hamiltonian, with fixed total angular momentum with respect to magnetic field axis. All three Hamiltonians describe N electrons and a fixed nucleus where the Coulomb interaction has been replaced by B-dependent one-dimensional effective (vector valued) potentials but without magnetic field. Two of them are solvable in at least the one electron case. We briefly sketch how these Hamiltonians can be used to analyze the bottom of the spectrum of such atoms.

H+2 in a strong magnetic field described via a solvable model

We consider the hydrogen molecular ion H+2 in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strength of the magnetic field. However, we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion can be obtained when incorporating perturbative corrections up to second order. Finally, we show that He3+2 exists for sufficiently large magnetic fields.

A one-dimensional model for many-electron atoms in extremely strong magnetic fields: maximum negative ionization

We consider a one-dimensional model for many-electron atoms in strong magnetic fields in which the Coulomb potential and interactions are replaced by one-dimensional regularizations associated with the lowest Landau level. For this model we show that the maximum number of electrons satisfies a bound of the form where Z denotes the charge of the nucleus, B the field strength and c is a constant. We follow Liebs strategy in which convexity plays a critical role. For the case N = 2 with fractional nuclear charge, we also discuss the critical value at which the nuclear charge becomes too weak to bind two electrons.

H2 molecule in strong magnetic fields

The Pauli Hamiltonian of a molecule with fixed nuclei in a strong constant magnetic field is asymptotic, in norm-resolvent sense, to an effective Hamiltonian which has the form of a multi-particle Schrodinger operator with interactions given by one-dimensional δ-potentials. We study this effective Hamiltonian in the case of the H2 molecule and establish the existence of the ground state. We also show that the inter-nuclear equilibrium distance tends to 0 as the field strength tends to infinity.

Asymptotic behaviour of the equilibrium nuclear separation for the H+2 molecule in a strong magnetic field

We consider the hydrogen molecular ion H+2 in the fixed nuclear approximation, in the presence of a strong homogeneous magnetic field. We determine the leading asymptotic behaviour for the equilibrium distance between the nuclei of this molecule in the limit when the strength of the magnetic field goes to infinity.

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Abstract We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.

One-dimensional models for atoms in strong magnetic fields, II: Anti-Symmetry in the Landau Levels ∗

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulomb potential and interactions are replaced by regularizations associated with the lowest Landau band. For a large class of models of this type, we show that the maximum number of electrons that can be bound is less than aZ+Zf(Z). The function f(Z) represents a small non-linear growth which reduces to ApZ(logZ)2when the magnetic field B=O(Zp) grows polynomially with the nuclear charge Z. In contrast to earlier work, the models considered here include those arising from realistic cases in which the full trial wave function for N-electrons is the product of an N-electron trial function in one-dimension and an antisymmetric product of states in the lowest Landau level.

Auto-tail Dependence Coefficients for Stationary Solutions of Linear Stochastic Recurrence Equations and for GARCH(1,1)

We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed.