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

We study a spectral problem $(\mathscr{P}^{\delta})$ for a diffusion like equation in a 3D domain $\Omega$. The main originality lies in the presence of a parameter $\sigma^{\delta}$, whose sign changes on $\Omega$, in the principal part of the operator we consider. More precisely, $\sigma^{\delta}$ is positive on $\Omega$ except in a small inclusion of size $\delta>0$. Because of the sign-change of $\sigma^{\delta}$, for all $\delta>0$ the spectrum of $(\mathscr{P}^{\delta})$ consists of two sequences converging to $\pm\infty$. However, at the limit $\delta=0$, the small inclusion vanishes so that there should only remain positive spectrum for $(\mathscr{P}^{\delta})$. What happens to the negative spectrum? In this paper, we prove that the positive spectrum of $(\mathscr{P}^{\delta})$ tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of $(\mathscr{P}^{\delta})$ behaves like $\delta^{-2}\mu$ for some constant $\mu<0$. 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.