量子散射计算中赝时薛定谔方程的进一步推广和数值实现
Further generalization and numerical implementation of pseudo-time Schroedinger equations for quantum scattering calculations
摘要：
我们回顾和进一步发展最近推出的散射计算的数值方法的基础上，所谓的伪时间薛定谔方程，这又是一个修改的阻尼Chebyshev多项式展开计划。该方法利用了与时间无关的Schroedinger方程中吸收势的一种特殊的能量依赖形式，其中复能谱E_k在单位圆盘内映射到u_k，其中u_k是某个显式已知的稀疏矩阵U的特征值。最重要的是，对于数值实现，所有的物理特征值u_k是U的极端特征值，这允许人们使用滤波器对角化方法通过伪时间自相关函数的谐波求逆来非常有效地提取这些特征值。自相关函数的2T步的计算仅需要T个稀疏实矩阵向量乘法。我们描述和比较不同的计划，有效地对应于不同的选择的能量依赖的吸收潜力，并测试它们的数值计算共振的HCO分子。我们的数值试验提出了一个最佳的方案，提供准确的估计大多数共振态。
英文摘要：
We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme. The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schroedinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U. Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications. We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states.
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