具有连续谱的PT对称势
PT-symmetric potentials having continuous spectra
摘要：
研究了具有连续谱的一维PT对称量子力学哈密顿量。所考虑的哈密顿量具有$H=p^2+V(x)$的形式，其中$V(x)$在$x$中是奇数，纯虚的，并且作为$|x|\to\infty$而消失。五个PT对称的潜力进行了研究：Scarf-II潜在的$V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)$，大$|x|$指数衰减;合理的潜力$V_2(x)=iA_2\,x/(1+x^4)$和$V_3(x)=iA_3\,x/(1+|x|^3)$，大$|x|$代数衰减;阶跃函数潜在的$V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)$，它有紧凑的支持;调节库仑电位$V_5(x)=iA_5\,x/(1+x^2)$，它衰减缓慢，$|x|\to\infty$，可被视为一个远程电位。实数参数$A_n$测量这些势的强度。数值技术解决与这些潜在的时间无关的薛定谔本征值问题表明，相应的哈密顿量的频谱表现出普遍的性质。一般来说，特征值是部分实的，部分复的。实本征值形成频谱的连续部分，而复本征值形成频谱的离散部分。实本征值的范围从$0$到$+\infty$连续变化。复本征值出现在离散的复共轭对和$V_n(x)$（$1\leq n\leq4$）的这些对的数量是有限的，并随着强度参数$A_n$的值的增加而增加。然而，对于$V_5(x)$有一个{ \it无限}序列的离散特征值的极限点在原点。这个序列是复杂的，但它类似于氢原子的巴尔末级数，因为它具有平方反比收敛。
英文摘要：
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five PT-symmetric potentials are studied: the Scarf-II potential $V_1(x)=iA_1\,{\rm sech}(x)\tanh(x)$, which decays exponentially for large $|x|$; the rational potentials $V_2(x)=iA_2\,x/(1+x^4)$ and $V_3(x)=iA_3\,x/(1+|x|^3)$, which decay algebraically for large $|x|$; the step-function potential $V_4(x)=iA_4\,{\rm sgn}(x)θ(2.5-|x|)$, which has compact support; the regulated Coulomb potential $V_5(x)=iA_5\,x/(1+x^2)$, which decays slowly as $|x|\to\infty$ and may be viewed as a long-range potential. The real parameters $A_n$ measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from $0$ to $+\infty$. The complex eigenvalues occur in discrete complex-conjugate pairs and for $V_n(x)$ ($1\leq n\leq4$) the number of these pairs is finite and increases as the value of the strength parameter $A_n$ increases. However, for $V_5(x)$ there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
