AI 中文总结
本文针对全纯函数零点分布建立复分析结果，将其应用于奇数维欧氏空间薛定谔算子，证明一维类跳跃奇点势可产生大量甚至无穷多串共振，并关联其性质与势参数及散射行列式渐近行为。
AI 中文摘要
受散射理论的启发，本文证明了关于闭下半平面内某类全纯函数在实轴对数邻域内零点分布的一般性结果。我们定义了一种新的“指示函数”，用于衡量全纯函数沿对数曲线的增长性，并将其与该函数的零点分布建立关联，这与完全正则增长整函数在扇形区域内零点分布的经典结果相关。这些结果可应用于奇数维欧氏空间薛定谔算子的散射矩阵行列式，从而得到实轴对数邻域内共振计数函数的界。作为进一步应用，我们研究了一维具有类跳跃奇点的某类势函数，利用复分析结果证明，此类奇点可沿对数曲线产生大量——甚至无穷多串共振；我们将这些共振串的性质（包括位置和线性密度）与描述势函数奇点的参数、以及散射行列式的渐近行为建立关联。
英文摘要
Motivated by scattering theory, this paper proves general results about the distribution of zeros in logarithmic neighborhoods of the real axis for a certain class of functions holomorphic in the closed lower half-plane. We define a new \emph{indicator function} that measures the growth of the holomorphic function along logarithmic curves, and connect this to the distribution of zeros of the function. This is related to classical results on the distribution of zeros in sectors for entire functions of completely regular growth. These results can be applied to the determinant of the scattering matrix of a Schrodinger operator on odd-dimensional Euclidean space, yielding bounds on resonance counting functions for logarithmic neighborhoods of the real axis. As a further application, we study a certain family of potentials in one dimension having jump-like singularities. Using our complex-analytic results we show that the singularities can lead to many -- and even infinitely many -- strings of resonances along logarithmic curves. We connect the properties of these strings of resonances, including their location and linear density, both to the parameters describing the singularities of the potential and to the asymptotic behavior of the scattering determinant.
Comments29 pages, 0 figures
