arXiv 1604.035742016-04-12hep-th

经典不规则块体、Hill方程和PT对称周期复势

Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials

Marcin Piatek, Artur R. Pietrykowski

备注：: 23 pages, 1 figure
DOI：: 10.1007/JHEP07(2016)131

摘要：

利用二维共形场论（2dCFT）中Whittaker-Hill势$Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4hμ\cos2x$和周期复势$Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$的实现，研究了它们的Schroedinger本征值问题.结果表明，对于弱耦合（小）$h\in\mathbb{R}$和非整数Floquet参数的哈密顿量$H_{i}\!=\!-{\rm d}^2/{\rm d}x^2 + Q_{i}(x)$的$ν\notin\mathbb{Z}$谱，$i=1,2$和相应的两个线性无关的本征函数由“单味”和“双味”（$N_f=1,2$）不规则共形块的经典极限给出.众所周知，复杂的非厄米哈密顿是PT对称的（=在同时宇称P和时间反演T变换下不变），可以有实特征值。对于$h,x\in\mathbb{R}$，哈密顿$H_{1}$是PT对称的。结果表明，在弱耦合区，对于$ν\in\mathbb{R}\setminus\mathbb{Z}$，$H_{1}$具有实谱.这一事实在一个基本的方式遵循的$N_f=1$经典不规则块的定义。因此，$H_{1}$可以作为另一个新的模型来检验PT对称量子力学的假设。

英文摘要：

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4hμ\cos2x$ and the periodic complex potential $Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$ are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) $h\in\mathbb{R}$ and non-integer Floquet parameter $ν\notin\mathbb{Z}$ spectra of hamiltonians $H_{i}\!=\!-{\rm d}^2/{\rm d}x^2 + Q_{i}(x)$, $i=1,2$ and corresponding two linearly independent eigenfunctions are given by the classical limit of the "single flavor" and "two flavors" ($N_f=1,2$) irregular conformal blocks. It is known that complex non-hermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian $H_{1}$ is PT-symmetric for $h,x\in\mathbb{R}$. It is found that $H_{1}$ has a real spectrum in the weak coupling region for $ν\in\mathbb{R}\setminus\mathbb{Z}$. This fact in an elementary way follows from a definition of the $N_f=1$ classical irregular block. Thus, $H_{1}$ can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.

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