楔形轮廓上PT对称问题的算子解法
Operator based approach to PT-symmetric problems on a wedge-shaped contour
摘要：
本文考虑一类二阶微分方程$$ -y''(z)-(iz)^{N+2}y(z)=λy(z), \quad z\in Γ$$，其特征参数为$λ\in \mathbb{C}$ .在$\mathcal{PT}$量子力学中，$z$穿过一个复杂的轮廓$Γ\subset \mathbb{C}$，它通常不是实直线，也不是实半直线。通过参数化，我们将问题映射到实数线上，得到了$[0,\infty)$和$(-\infty,0].$上的两个微分方程，它们通过边界条件在零处耦合，并且它们的势不是实数值。主要结果是沿着著名的极限点/极限圆方案介绍的复势的分类这一问题。60年前的西姆斯此外，我们将算子与两个半直线问题和全轴问题联系起来，研究了它们的谱。
英文摘要：
We consider a second-order differential equation $$ -y''(z)-(iz)^{N+2}y(z)=λy(z), \quad z\in Γ$$ with an eigenvalue parameter $λ\in \mathbb{C}$. In $\mathcal{PT}$ quantum mechanics $z$ runs through a complex contour $Γ\subset \mathbb{C}$, which is in general not the real line nor a real half-line. Via a parametrization we map the problem back to the real line and obtain two differential equations on $[0,\infty)$ and on $(-\infty,0].$ They are coupled in zero by boundary conditions and their potentials are not real-valued. The main result is a classification of this problem along the well-known limit-point/ limit-circle scheme for complex potentials introduced by A.R.\ Sims 60 years ago. Moreover, we associate operators to the two half-line problems and to the full axis problem and study their spectra.
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