A negative result for hearing the shape of a triangle: a computer-assisted proof - Orriols Giménez, Gerard (Reports@SCM, 2020)
We prove that there exist two distinct triangles for which the rst, second and fourth eigenvalues of the Laplace operator with zero Dirichlet boundary conditions coincide. This solves a conjecture raised by Antunes and Freitas and suggested by their numerical evidence. We use a novel technique for a computer-assisted proof about the spectrum of an operator, which combines a Finite Element Method, to locate roughly the rst eigenvalues keeping track of their position in the spectrum, and the Method of Particular Solutions, to get a much more precise bound of these eigenvalues. Keywords: computer-assisted proof, Laplace eigenvalues, spectral geometry, Finite Element Method, Method of Particular Solutions.