Uniform spectral approximation for regular Sturm–Liouville operators

Establish rigorously that discrete sine transform based discretisation uniformly approximates the first N eigenvalues of regular Sturm–Liouville operators in Liouville normal form with zero Dirichlet boundary conditions, including both low- and high-index eigenvalues.

Background

The paper observes that high-index eigenvalues of regular Sturm–Liouville operators in Schrödinger form have the same leading asymptotic behaviour as those of the second-derivative operator with zero Dirichlet boundary conditions. Because the discrete sine transform exactly reproduces the latter spectrum, the authors propose using it to obtain accurate approximations across the entire spectrum rather than only for low-index eigenvalues.

The claim is presented as a conjecture and is tested numerically on benchmark Paine and Coffey–Evans problems. The numerical evidence supports the conjecture, but the paper does not provide a rigorous convergence or uniform-error theorem establishing it in general.

References

As conjectured by Fusi et al. (2026), the discrete sine transform based discretisation might still have an edge regarding the uniform approximation of spectra even in complex settings beyond the simple eigenvalue problem for the second derivative operator (1.4).

— Uniform approximation of spectra of linear second order differential operators via discrete sine transform based discretisation  (2609.25796 - Křenek et al., 22 Sep 2026) in Section 1.4, page 3

Consequently, we can conjecture that the discrete sine transform based discretisation should recover the asymptotic behaviour (2.13), and hence that it is convenient for the approximation of high-index eigenvalues.

— Uniform approximation of spectra of linear second order differential operators via discrete sine transform based discretisation  (2609.25796 - Křenek et al., 22 Sep 2026) in Section 2.2, page 7