We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1,\infty) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ<1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a<(1-λ^2)^{1/2}, for λ\in[0,1), and a<1, for λ<0.
New approach to numerical computation of the eigenfunctions of the continuous spectrum of three-particle Schrödinger operator. I. One-dimensional particles, short-range pair potentials
On Hilbert-Polya conjecture: Hermitian operator naturally associated to L-functions
Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero fo Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is can…
Vector orthogonal polynomials with Bochner's property
Jacobi、Hermite、Laguerre和Bessel等经典的正交多项式系统具有多项式是二阶普通微分算子的本征函数的性质。根据Bochner的一个经典定理，它们是唯一具有这种性质的系统。类似地，Charlier、Meixner、Kravchuk和Hahn的多项式都是正交的，并且是适当的二阶差分算子的本征函数。根据著名的Favard-Shohat定理，正交性条件等价于3项递归关系。向量正交多项式（VOP）满足有限项递归关系，其中有更多的条款，根据一个定理，由J. Van Iseghem，这是他们的特点。受Bochner定理的启发，我们正在寻找VOP，它也是微分（差分）算子的本征函数。我们称这些同时条件为Bochner性质。本文的目的是介绍具有Bochner性质的VOP的构造方法。这些方法是纯代数的，并且基于非交换代数的自同构。他们还使用了所谓的双频问题的想法。抽象方法的应用包括经典正交多项式的广泛推广，包括连续和离散。其他结果连接不同的家庭VOP，包括经典的，由纯代数起源的线性变换，尽管事实上，当解释分析，他们是积分变换。
