具有一般色散关系的普适散射
Universal scattering with general dispersion relations
摘要：
许多合成量子系统允许粒子具有既不是线性函数也不是二次函数的色散关系。在这里，我们探讨一般空间维度$D\geq 1$的单粒子散射时，在一个特定的能量状态的密度发散。为了说明在实验相关的设置的基本原理，我们专注于波导量子电动力学（QED）问题（即$D=1$）与色散关系$ε(k)=\pm |d|k^m$，其中$m\geq 2$是一个整数。对于任何正整数$m$的这些问题的一大类，我们严格证明，当没有明亮的零能量本征态，$S$矩阵评估的能量$E\to 0$收敛到一个普遍的限制，仅依赖于$m$。我们还给出了一个概括的关键指数定理在量子散射理论被称为莱文森定理-这涉及到的散射相位的束缚态的数量-波导QED散射这些更一般的色散关系。然后，我们将这些结果扩展到一般的整数维$D \geq 1$，色散关系$ε(\boldsymbol{k}) = |\boldsymbol{k}|^a$的$D$维动量矢量$\boldsymbol{k}$与任何真正的积极$a$，和可分离的潜在散射。
英文摘要：
Many synthetic quantum systems allow particles to have dispersion relations that are neither linear nor quadratic functions. Here, we explore single-particle scattering in general spatial dimension $D\geq 1$ when the density of states diverges at a specific energy. To illustrate the underlying principles in an experimentally relevant setting, we focus on waveguide quantum electrodynamics (QED) problems (i.e. $D=1$) with dispersion relation $ε(k)=\pm |d|k^m$, where $m\geq 2$ is an integer. For a large class of these problems for any positive integer $m$, we rigorously prove that when there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an energy $E\to 0$ converges to a universal limit that is only dependent on $m$. We also give a generalization of a key index theorem in quantum scattering theory known as Levinson's theorem -- which relates the scattering phases to the number of bound states -- to waveguide QED scattering for these more general dispersion relations. We then extend these results to general integer dimensions $D \geq 1$, dispersion relations $ε(\boldsymbol{k}) = |\boldsymbol{k}|^a$ for a $D$-dimensional momentum vector $\boldsymbol{k}$ with any real positive $a$, and separable potential scattering.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
