特殊点上的射影Hilbert空间结构
Projective Hilbert space structures at exceptional points
摘要：
利用非Hermitian复对称2 × 2矩阵玩具模型研究了例外点附近的射影Hilbert空间结构。对可对角化矩阵的双正交特征向量，用EP上的根向量进行了Pui-pix展开。结果表明，在EP极限中具有有限性和奇异性的两个不相容的规范化条件之间的矛盾可以通过对原Hilbert空间的射影扩展来解决.互补归一化条件则对应于这个扩大的射影希尔伯特空间的两个不同的仿射图。几何相位和相位跳跃行为进行了分析和有用的相位刚度作为衡量EP配置的距离。最后，讨论了PT对称扩展量子力学中与EP有关的方面，并提出了一个关于量子最速线问题的猜想。
英文摘要：
A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.
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