论文标题
具有连续光谱的PT对称电势
PT-symmetric potentials having continuous spectra
论文作者
论文摘要
研究了具有连续光谱的一维PT对称量子力学的哈密顿量。被认为的汉密尔顿人具有$ h = p^2+v（x）$的形式，其中$ v（x）$在$ x $，纯虚构中是奇怪的，并且消失为$ | x | \ to \ to \ infty $。研究了五个PT-对称电位：围巾-II电位$ v_1（x）= ia_1 \，{\ rm sech}（x）\ tanh（x）$，它对大的$ | x | $呈指数型衰减；理性电势$ v_2（x）= ia_2 \，x/（1+x^4）$和$ v_3（x）= ia_3 \，x/（1+ | x |^3）$，对于大$ | x | $而言，该代数的衰减为代数。具有紧凑的支撑的步骤功能电位$ v_4（x）= ia_4 \，{\ rm sgn}（x）θ（2.5- | x |）$；受调节的库仑电势$ v_5（x）= ia_5 \，x/（1+x^2）$，其衰减缓慢，损失为$ | x | \ to \ infty $，并且可能被视为远距离电位。实际参数$ a_n $衡量这些电位的优势。解决与这些电势相关的时间独立的Schrödinger特征值问题的数值技术表明，相应的汉密尔顿人的光谱表现出通用性能。通常，特征值部分是真实的，部分是复杂的。实际特征值形成了光谱的连续部分，并且复杂的特征值形成了光谱的离散部分。真正的特征值从$ 0 $到$+\ infty $持续不断范围。复杂的特征值出现在离散的复杂偶联对，对于$ v_n（x）$（$ 1 \ leq n \ leq4 $），这些对的数量是有限的，并且随着强度参数$ a_n $的值的增加而增加。但是，对于$ v_5（x）$，有一个{\ it iT Infinite}的离散特征值序列，其原点处有一个极限点。该序列很复杂，但与Balmer系列的氢原子相似，因为它具有反方面的收敛。
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.
