PDMATHAAB哈密顿量${\cal P}{\cal T}$对称破缺解的多尺度参考函数分析
Multiscale Reference Function Analysis of the ${\cal P}{\cal T}$ Symmetry Breaking Solutions for the $P^2+iX^3+iαX$ Hamiltonian
摘要：
Delabaere和Trinh（2000 J. Phys. A 33 8771）最近的工作发现了一维哈密顿量$P^2+iX^3+iαX$在渐近极限$α\to -\infty$中存在${\cal P}{\cal T}$ -对称破缺、复能量、$L^2$解。他们的渐近分析对$α$的中等值产生了可疑的结果。我们可以很容易地确认存在${\cal P}{\cal T}$对称性破缺解决方案，明确计算的低躺状态，$|α| < O (10)$。我们的分析利用了多尺度参考函数（MRF）方法，由Tymczak等人（1998年物理评论快报）开发。80 3678; 1998 Phys.Rev.A58，2708）。MRF的结果可以通过与Handy（2001 a，b）最近提出的特征值矩方法产生的收敛特征能边界进行比较来验证。由于MRF分析的可靠性，其快速的数值实现，高精度，和理论简单，本形式主义定义了一个有效的和高效的程序，用于分析最近的文献中出现的许多相关问题。
英文摘要：
The recent work of Delabaere and Trinh (2000 J. Phys. A 33 8771) discovered the existence of ${\cal P}{\cal T}$-symmetry breaking, complex energy, $L^2$ solutions for the one dimensional Hamiltonian, $P^2+iX^3+iαX$, in the asymptotic limit, $α\to -\infty$. Their asymptotic analysis produced questionable results for moderate values of $α$. We can easily confirm the existence of ${\cal P}{\cal T}$-symmetry breaking solutions, by explicitly computing the low lying states, for $|α| < O (10)$. Our analysis makes use of the Multiscale Reference Function (MRF) approach, developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3678; 1998 Phys. Rev. A 58, 2708). The MRF results can be validated by comparing them with the converging eigenenergy bounds generated through the Eigenvalue Moment Method, as recently argued by Handy (2001a,b). Given the reliability of the MRF analysis, its fast numerical implementation, high accuracy, and theoretical simplicity, the present formalism defines an effective and efficient procedure for analyzing many related problems that have appeared in the recent literature.
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