$\mathbb R^3$中不定常Schrödinger型方程的吸收边界条件
Absorbing boundary conditions for the time-dependent Schrödinger-type equations in $\mathbb R^3$
摘要：
提出了三维不定常薛定谔型方程的吸收边界条件，作为减少量子力学计算量的一种手段.边界条件首先从薛定谔方程的半离散逼近导出，其优点是所得公式自动与有限差分格式兼容，并且不需要在空间中进一步离散。吸收边界条件表示为离散的狄利克雷-诺依曼（Dirichlet-to-Neumann，DtN）映射，其可以通过使用拉普拉斯变换的有理逼近在时间上进一步近似，以实现更有效的实现。这种方法可以应用于具有任意几何形状的域。证明了零阶和一阶吸收边界条件的稳定性。我们在基准问题上测试了边界条件。通过一个含Skyrme相互作用的不定常Hartree-Fock模型进一步验证了该方法的有效性。能量和核子密度方面的准确性进行了检查。
英文摘要：
Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semi-discrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann (DtN) map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth order and first order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.
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