标题： 克雷因矩阵：一般理论及在原子玻色-爱因斯坦凝聚体中的具体应用 显示英文标题

标题： The Krein Matrix: General Theory and Concrete Applications in Atomic Bose-Einstein Condensates

摘要： 在寻找哈密顿本征值问题的非零本征值时，特别重要的是不仅要确定不稳定的本征值（即实部为正的本征值），还要确定那些纯虚数但具有负Krein符号的本征值。 这些后者的本征值具有这样的性质，它们在与其他纯虚数本征值发生碰撞时可能会变得不稳定，即它们是导致所谓的哈密顿-霍普夫分支机制的必要组成部分。 在本文中，我们回顾了一种一般理论，用于构造一个全纯的矩阵值函数，即所谓的Krein矩阵，该函数不仅具有定位不稳定本征值的性质，还具有定位具有负Krein符号的本征值的性质。 这些本征值被实现为行列式的零点。 通过将Krein矩阵的行列式设为零得到的有限维问题提供了一个有价值的简化。 在本文中，通过最近在原子玻色-爱因斯坦凝聚体实验和理论研究中出现的状态的谱分析典型例子来说明该技术的有用性。 特别是，我们考虑了一维设置（香肠陷阱）具有实值多暗孤子解，以及二维设置（薄饼陷阱）允许复值多涡旋定态波形。 显示英文摘要

摘要： When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part), but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues, i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable eigenvalues, but also those with negative Krein signature. These eigenvalues are realized as zeros of the determinant. The resulting finite dimensional problem obtained by setting the determinant of the Krein matrix to zero presents a valuable simplification. In this paper the usefulness of the technique is illustrated through prototypical examples of spectral analysis of states that have arisen in recent experimental and theoretical studies of atomic Bose-Einstein condensates. In particular, we consider one-dimensional settings (the cigar trap) possessing real-valued multi-dark-soliton solutions, and two-dimensional settings (the pancake trap) admitting complex multi-vortex stationary waveforms.