标题： FEM-DtN-SIM方法计算薛定谔算子的共振 显示英文标题

标题： FEM-DtN-SIM Method for Computing Resonances of Schrödinger Operators

摘要： 薛定谔算子的共振研究在数学物理中有着悠久的传统。大量的理论研究已经探讨了共振接近实轴的程度、它们的分布以及计数函数的界。然而，由于问题的非线性和区域的无界性，一维以外的计算结果仍然很少见。我们提出了一种新的方法，该方法集成了有限元法、Dirichlet到Neumann（DtN）映射和谱指标法。在截断计算域边界上施加的DtN映射强制执行发散条件。有限元法允许有效地处理复杂的势函数。谱指标法能够有效计算（复数）特征值，而不引入谱污染。这种方法的有效性通过一系列数值例子得到了验证。 显示英文摘要

摘要： The study of resonances of the Schr\"{o}dinger operator has a long-standing tradition in mathematical physics. Extensive theoretical investigations have explored the proximity of resonances to the real axis, their distribution, and bounds on the counting functions. However, computational results beyond one dimension remain scarce due to the nonlinearity of the problem and the unbounded nature of the domain. We propose a novel approach that integrates finite elements, Dirichlet-to-Neumann (DtN) mapping, and the spectral indicator method. The DtN mapping, imposed on the boundary of a truncated computational domain, enforces the outgoing condition. Finite elements allow for the efficient handling of complicated potential functions. The spectral indicator method effectively computes (complex) eigenvalues of the resulting nonlinear algebraic system without introducing spectral pollution. The viability of this approach is demonstrated through a range of numerical examples.