2025/04/16 by Bo Gong, Takumi Sato, Gong, Bo +5
Computer Science · Mathematics · Physics and Astronomy · #65H17 #65N25 #78M10 #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2504.11680
openalex publication_date 2025/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The study of resonances of the Schrödinger 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.