Boundedness of Left Half-Plane Eigenvalues for Non-Selfadjoint Indefinite Sturm--Liouville Problems with Applications to Fourier Modal Methods

We study a class of Sturm–Liouville problems of the form $$-(p{y}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+qy=\lambda py,$$ on a finite interval with complex-valued coefficients, where $p$ is piecewise smooth 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. A canonical instance of this class arises in transverse-magnetic ™ diffraction by metallic lamellar gratings, a benchmark problem in computational photonics that has been central to the development of Fourier modal methods. These methods exhibit long-standing convergence difficulties in this setting, associated with the loss of definiteness of the underlying operator and the emergence of spurious modes. Our result yields a rigorous criterion for identifying such non-physical modes in low-loss metallic gratings. Numerical examples illustrate the practical utility of the criterion.