scholarly journals CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

2000 ◽  
Vol 12 (02) ◽  
pp. 181-225 ◽  
Author(s):  
KURT BRODERIX ◽  
DIRK HUNDERTMARK ◽  
HAJO LESCHKE
Keyword(s):  
Brownian Bridge ◽  
Integral Kernel ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Kato Class ◽  
Vector Potentials ◽  
Continuity Properties ◽  
Scalar Potentials ◽  
General Configuration ◽  
Zero Vector

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.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

1989 ◽  
Vol 21 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. R. Lerche ◽  
D. Siegmund
Keyword(s):  
Virial Coefficient ◽  
Large Deviation ◽  
Brownian Bridge ◽  
Exit Time ◽  
Second Virial Coefficient ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Short Time Interval ◽  
First Exit Time ◽  
Short Time

Let T be the first exit time of Brownian motion W(t) from a region ℛ in d-dimensional Euclidean space having a smooth boundary. Given points ξ0 and ξ1 in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε |W(0) = ξ0, W(ε) = ξ 1} as ε → 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

scholarly journals Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

2003 ◽  
Vol 2003 (12) ◽  
pp. 715-741 ◽  
Author(s):  
Imed Bachar ◽  
Habib Màagli ◽  
Syrine Masmoudi ◽  
Malek Zribi
Keyword(s):  
Green Function ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Singular Solutions ◽  
Kato Class ◽  
New Class ◽  
Nonlinear Elliptic ◽  
New Form ◽  
The Green Function ◽  
Class Of Functions

We establish a new form of the3Gtheorem for polyharmonic Green function on the unit ball ofℝn(n≥2)corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functionsKm,ncontaining properly the classical Kato classKn. We exploit properties of functions belonging toKm,nto prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order2m.

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

scholarly journals Continuity properties of the semi-group and its integral kernel in non-relativistic QED

2016 ◽  
Vol 28 (05) ◽  
pp. 1650011 ◽  
Author(s):  
Oliver Matte
Keyword(s):  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Integral Kernel ◽  
Model Parameters ◽  
Semi Group ◽  
Continuity Properties ◽  
The Standard Model ◽  
Spatial Coordinates ◽  
Spectral Subspaces ◽  
Vector Valued

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. Güneysu, J. S. Møller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued [Formula: see text]-spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary, we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected, we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter.

The Potentials

2019 ◽  
pp. 43-60
Author(s):  
Richard Freeman ◽  
James King ◽  
Gregory Lafyatis
Keyword(s):  
Wave Equations ◽  
Electric Quadrupole ◽  
Quadrupole Moments ◽  
Retarded Time ◽  
Vector Potentials ◽  
Electric And Magnetic Fields ◽  
Lorenz Gauge ◽  
The One ◽  
Scalar Potentials ◽  
Approximate Expressions

The concepts of scalar and vector potentials are introduced and the electric and magnetic fields are shown to be derived from specific forms of these potentials. The choice of these forms is restricted by gauge considerations, and the Lorenz gauge is introduced as the one most applicable for radiation. Using this, the wave equations prescribing the potentials in terms of the source conditions are presented. The modifications of vector and scalar potentials to account for speed of light and causality lead to the concept of “retarded time.” The potentials can be expressed in terms of moments of the source along with concepts of “near,” “intermediate,” and “far” zones to facilitate derivation of approximate expressions for the potentials evaluated at appropriate distances from the source. Finally, expressions for the vector potential in terms of the electric and magnetic dipole, and electric quadrupole moments of the source in the approximation zones are presented.

Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

1989 ◽  
Vol 21 (01) ◽  
pp. 1-19 ◽  
Author(s):  
H. R. Lerche ◽  
D. Siegmund
Keyword(s):  
Virial Coefficient ◽  
Large Deviation ◽  
Brownian Bridge ◽  
Exit Time ◽  
Second Virial Coefficient ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Short Time Interval ◽  
First Exit Time ◽  
Short Time

LetTbe the first exit time of Brownian motionW(t) from a region ℛ ind-dimensional Euclidean space having a smooth boundary. Given points ξ0and ξ1in ℛ, ordinary and large-deviation approximations are given for Pr{T &lt; ε|W(0) = ξ0,W(ε)=ξ1} asε→ 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

scholarly journals Acceleration of Augmented EFIE Using Multilevel Complex Source Beam Method

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Lianning Song ◽  
Yongpin Chen ◽  
Ming Jiang ◽  
Jun Hu ◽  
Zaiping Nie
Keyword(s):  
Computational Cost ◽  
Low Frequency ◽  
Electric Field Integral Equation ◽  
Vector Potentials ◽  
Multilevel Algorithm ◽  
Complex Source ◽  
The Matrix ◽  
Aggregation And Disaggregation ◽  
Scalar Potentials ◽  
Fast Multipole Algorithm

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.

scholarly journals Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

2006 ◽  
Vol 4 (4) ◽  
Author(s):  
Daniel Boykis ◽  
Patrick Moylan
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flat Space ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Curved Space

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.

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