Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

We study a spectral problem (P^δ) for a diffusion-like equation in a 3D domain Ω. The main originality lies in the presence of a parameter σ^δ, whose sign changes on Ω, in the principal part of the operator we consider. More precisely, σ^δ is positive on Ω except in a small inclusion of size δ>0. Because of the sign change of σ^δ, for all δ>0, the spectrum of (P^δ) consists of two sequences converging to ±∞. However, at the limit δ=0, the small inclusion vanishes so that there should only remain positive spectrum for (P^δ). What happens to the negative spectrum? In this paper, we prove that the positive spectrum of (P^δ) tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of (P^δ) behaves like δ^-2μ for some constant μ<0. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.