FEM-DtN-SIM method for computing resonances of Schrödinger operators

Document Type

Article

Publication Date

2-1-2026

Abstract

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. Finally, the spectral indicator method is employed to compute (complex) eigenvalues of the resulting nonlinear algebraic system. The viability of this approach is demonstrated through a range of numerical examples.

Publication Title

Applied Mathematics Letters

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