On a conjecture of Bennewitz, and the behaviour of the Titchmarsh-Weyl matrix near a pole
B.M.Brown and M. Marletta
对于 [0,∞] 上的任何真正的 limit-n 2n 阶自相邻线性差分表达式,可以定义 Titchmarsh- Weyl matrices M(λ)。 particu lar interest的两个矩阵是矩阵M_D(λ)和M_N(λ) assoc,分别与Dirichlet和Neumann边界条件在x=0。 这些满足M_D(λ) = -M_N(λ)^-1。 众所周知,当这些矩阵有极点(只能位于真实轴上)时,有效的帮助不平等的存在取决于它们在这些极点附近的行为。 我们证明了Bennewitz的一个猜想,并使用它,连同一个新的算法,用于计算Titchmarsh-Weyl矩阵在极点附近的Laurent扩展,以调查HELP不等式是否存在,因为一些微分方程迄今为止证明是笨拙的分析。
For any real limit-n 2nth-order selfadjoint linear differential expression on [0,∞), Titchmarsh- Weyl matrices M(λ) can be defined. Two matrices of particu lar interest are the matrices M_D(λ) and M_N(λ) assoc iated respectively with Dirichlet and Neumann boundary conditions at x=0. These satisfy M_D(λ) = -M_N(λ)^-1. It is known that when these matrices have poles (which can only lie on the real axis) the existence of valid HELP inequalities depends on their behaviour in the neighbourhood of the...