The purpose of this talk is to investigate the spectral effects of an interface between a usual dielectric and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation (which actually highlights most spectral features of more general situations), namely: 1) the simplest existing model of NIM : the Drude model (for which negativity occurs at low frequencies), 2) a two-dimensional scalar model derived from the complete Maxwell's equations, 3) the case of a simple bounded cavity : a camembert-like domain partially filled with a portion of non dissipative Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of additional unknowns, we will show how to linearize the problem and we will present a complete description of the spectrum. We will see in particular that, contrarily to the case of a cavity filled by a usual dielectric (for which the spectrum is always purely discrete), the presence of the interface of the Drude material is responsible for a component of essential spectrum, which contains an interval (of non-zero length) as soon as the interface contains a corner. This surprising component is related to a black-hole effect at the corner.