Boundedness of Left Half-Plane Eigenvalues for Coefficient-Coupled Sturm--Liouville Problems with Application to Fourier Modal Methods

Abstract

We study a class of Sturm--Liouville problems of the form \[ -(p\,y')' + q\,y = λ\, w\, y, \] on a finite interval with complex-valued coefficients, where $p$ and $w$ are piecewise smooth, the ratio $w/p$ is real and positive, and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which implies that only finitely many eigenvalues lie in this region. This stands in contrast to the known unboundedness, for real-valued coefficients, when $p$ or $w$ changes sign independently. A canonical instance of this class, with $w=p$, arises in transverse-magnetic (TM) diffraction by metallic lamellar gratings, a benchmark problem in computational photonics, central to the development of modal methods. In Fourier modal methods, in particular, the emergence of spurious modes with unbounded propagation constants (eigenvalues), rooted in discretization of sign-changing coefficients, leads to notorious convergence difficulties. These modes cannot be excluded \textit{a priori}, since genuine eigenvalues are not constrained by conventional bounds in this regime. Nevertheless, our result shows that the physical eigenvalues remain bounded, providing a rigorous criterion for identifying spurious modes in low-loss metallic gratings.

28 pages, 10 figures (V3:generalized to coefficient-coupled problems with w/p>0; proof simplified; explicit bound added)