Eigenvalue Falls in Thin Broken Quantum Strips

We are interested in the spectrum of the Dirichlet Laplacian in thin broken strips with angle [Formula: see text]. Playing with symmetries, this leads us to investigate spectral problems for the Laplace operator with mixed boundary conditions in trapezoids of thickness [Formula: see text] small. We give an asymptotic expansion of the first eigenvalues and corresponding eigenfunctions as [Formula: see text] tends to zero. The new point in this work is to study the dependence with respect to [Formula: see text]. We highlight a curious phenomenon of diving eigenvalues: when the strip is more and more broken, at certain critical angles, which we characterize, an eigenvalue moves down very rapidly below the pack of other eigenvalues. We prove that this occurs more gently at [Formula: see text] than at positive critical angles.