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Papers by Raymond Brummelhuis

Research paper thumbnail of Functionals of exponential Brownian motion and divided differences

Fuel and Energy Abstracts, 2011

We calculate an analytic value for the correlation coefficient beween a geometric, or exponential... more We calculate an analytic value for the correlation coefficient beween a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

Research paper thumbnail of One Dimensional Regularizations of the Coulomb Potential with Application to Atoms in Strong Magnetic Fields

We consider one-dimensional regularizations of the Coulomb potential formed by taking a two-dimen... more We consider one-dimensional regularizations of the Coulomb potential formed by taking a two-dimensional expectation of the Coulomb potential with respect to the Landau states. It is well-known that such functions arises naturally in the study of atoms in strong magnetic fields. For many-electron atoms consideration of the Pauli principle requires convex combinations of such potentials and interactions in which the regularizations also contain a 2^{-1/2} rescaling. We summarize the results of a comprehensive study of these functions including recursion relations, tight bounds, convexity properties, and connections with confluent hypergeometric functions. We also report briefly on their application in one-dimensional models of many-electrons atoms in strong magnetic fields.

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

Journal of Physics A-mathematical and General, 1999

We follow Lieb's strategy in which convexity plays a critical role. For the case of two electrons... more We follow Lieb's strategy in which convexity plays a critical role. For the case of two electrons and fractional nuclear charge, we also discuss the critical value at which the nuclear charge becomes too weak to bind two electrons.

Research paper thumbnail of One-Dimensional Models for Atoms in Strong Magnetic Fields, II: Anti-Symmetry in the Landau Levels

Journal of Statistical Physics, 2004

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulo... more 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 A p Z(logZ)2when the magnetic field B=O(Z p ) 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.

Research paper thumbnail of Exponential decay in the semi–classical limit for eigenfunctions of schrödinger operators with magnetic fields and potentials which degenerate at infinity

Communications in Partial Differential Equations, 1991

Research paper thumbnail of A necessary and sufficient condition for Melin’s inequality for a class of systems

Journal D Analyse Mathematique, 2001

Research paper thumbnail of Stationary Scattering in the Semiclassical Limit for a Schrödinger Operator in a Gauge Field

Journal of Functional Analysis, 1999

We study the classical limit of the stationary scattering theory for a Schro dinger operator in a... more We study the classical limit of the stationary scattering theory for a Schro dinger operator in a compactly supported gauge field. We show that, under suitable hypotheses on the associated classical flow, the scattering amplitude has a complete asymptotic expansion in the semi-classical parameter, and we determine the main term of this expansion.

Research paper thumbnail of Resonances of the Dirac Hamiltonian in the Non Relativistic Limit

Annales Henri Poincare, 2001

For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a... more For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a power of |x|, we prove the existence of resonances and we study, when \( c \rightarrow + \infty \) , the asymptotic expansion of their real part,and an estimation of their imaginary part, generalizing an old result of Titchmarsh.

Research paper thumbnail of Scattering amplitude for dirac operators

Communications in Partial Differential Equations, 1999

hl.Taylor has constructed an approximate diagonalization of a system of pseudodifferential operat... more hl.Taylor has constructed an approximate diagonalization of a system of pseudodifferential operators under the assumption that the matrix of principal symbols is diagonalizable in a C" way. Helffer and Sjostrand [9] have given the analogue of Taylor's construction in the semiclassical case. For some systems, like t h system of h9axwell equations ([13]), the hypothesis of global Cw diagonalizability cannot be met, for example for topological reasons. and it seems useful to have a variant of the above result which does not need this hypothesis, and gives approximate projectors rather than an approximate diagonalization. We present such a result in section 2 below, with a proof which differs from those in [17] and .

Research paper thumbnail of Functionals of exponential Brownian motion and divided differences

Fuel and Energy Abstracts, 2011

We calculate an analytic value for the correlation coefficient beween a geometric, or exponential... more We calculate an analytic value for the correlation coefficient beween a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

Research paper thumbnail of One Dimensional Regularizations of the Coulomb Potential with Application to Atoms in Strong Magnetic Fields

We consider one-dimensional regularizations of the Coulomb potential formed by taking a two-dimen... more We consider one-dimensional regularizations of the Coulomb potential formed by taking a two-dimensional expectation of the Coulomb potential with respect to the Landau states. It is well-known that such functions arises naturally in the study of atoms in strong magnetic fields. For many-electron atoms consideration of the Pauli principle requires convex combinations of such potentials and interactions in which the regularizations also contain a 2^{-1/2} rescaling. We summarize the results of a comprehensive study of these functions including recursion relations, tight bounds, convexity properties, and connections with confluent hypergeometric functions. We also report briefly on their application in one-dimensional models of many-electrons atoms in strong magnetic fields.

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

Journal of Physics A-mathematical and General, 1999

We follow Lieb's strategy in which convexity plays a critical role. For the case of two electrons... more We follow Lieb's strategy in which convexity plays a critical role. For the case of two electrons and fractional nuclear charge, we also discuss the critical value at which the nuclear charge becomes too weak to bind two electrons.

Research paper thumbnail of One-Dimensional Models for Atoms in Strong Magnetic Fields, II: Anti-Symmetry in the Landau Levels

Journal of Statistical Physics, 2004

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulo... more 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 A p Z(logZ)2when the magnetic field B=O(Z p ) 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.

Research paper thumbnail of Exponential decay in the semi–classical limit for eigenfunctions of schrödinger operators with magnetic fields and potentials which degenerate at infinity

Communications in Partial Differential Equations, 1991

Research paper thumbnail of A necessary and sufficient condition for Melin’s inequality for a class of systems

Journal D Analyse Mathematique, 2001

Research paper thumbnail of Stationary Scattering in the Semiclassical Limit for a Schrödinger Operator in a Gauge Field

Journal of Functional Analysis, 1999

We study the classical limit of the stationary scattering theory for a Schro dinger operator in a... more We study the classical limit of the stationary scattering theory for a Schro dinger operator in a compactly supported gauge field. We show that, under suitable hypotheses on the associated classical flow, the scattering amplitude has a complete asymptotic expansion in the semi-classical parameter, and we determine the main term of this expansion.

Research paper thumbnail of Resonances of the Dirac Hamiltonian in the Non Relativistic Limit

Annales Henri Poincare, 2001

For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a... more For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a power of |x|, we prove the existence of resonances and we study, when \( c \rightarrow + \infty \) , the asymptotic expansion of their real part,and an estimation of their imaginary part, generalizing an old result of Titchmarsh.

Research paper thumbnail of Scattering amplitude for dirac operators

Communications in Partial Differential Equations, 1999

hl.Taylor has constructed an approximate diagonalization of a system of pseudodifferential operat... more hl.Taylor has constructed an approximate diagonalization of a system of pseudodifferential operators under the assumption that the matrix of principal symbols is diagonalizable in a C" way. Helffer and Sjostrand [9] have given the analogue of Taylor's construction in the semiclassical case. For some systems, like t h system of h9axwell equations ([13]), the hypothesis of global Cw diagonalizability cannot be met, for example for topological reasons. and it seems useful to have a variant of the above result which does not need this hypothesis, and gives approximate projectors rather than an approximate diagonalization. We present such a result in section 2 below, with a proof which differs from those in [17] and .