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

TL;DR

This work tackles the challenge of computing resonances for Schrödinger operators in higher dimensions by reformulating the problem on a bounded domain using a Dirichlet-to-Neumann boundary map to enforce outgoing radiation. Finite element discretization yields a nonlinear eigenproblem for a holomorphic operator-valued function $F(k)$, which is solved efficiently with a parallel spectral indicator method based on contour integrals. The approach is validated with 2D numerical experiments showing accurate resonance detection and convergence under mesh refinement, with convergence behavior consistent with linear elements. The framework has broad applicability to scattering resonances in acoustic and electromagnetic settings and offers a path toward higher-dimensional computations.

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. 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.

Figures (3)

• Figure 1: Contour plots of $|d_n(k)|$, $n = 0, \ldots, 10$ ($r_0=1, V(x)=2$).
• Figure 2: Computed resonances of the ($r_0=1, V(x)=2$).
• Figure 3: Computed resonances in $\Theta$. First: Example 2. Second: Example 3. Third: Example 4. Fourth: Example 5.