CERAMATHS-DMATHS Seminar: Presentation by Hatem Najar

The CÉRAMATHS Department of Mathematics seminar series will host Hatem Najar (Faculty of Sciences, Monastir, Tunisia) at 2:00 p.m. on Thursday, July 2, 2026, for the following talk:

On the spectral theory of quantum waveguides with mixed boundary conditions

We consider the spectral analysis of the Laplace operator on a domain subject to mixed Dirichlet and Neumann boundary conditions. The objective is to analyze the influence of the coexistence of these two types of boundary conditions on the structure of the spectrum. After reviewing some classical results, we present new results in the case where the aperture is elliptical in shape. This geometry breaks rotational symmetry and introduces anisotropy along the semi-axes of the ellipse, thereby modifying the coupling of transverse modes as well as the low-energy spectrum. We show that the operator has a finite number of discrete eigenvalues located below the threshold of the essential spectrum, and we study their dependence on the geometric parameters of the ellipse. Finally, numerical simulations highlight the evolution of the first eigenvalues and the ground state as a function of the window geometry.