Can one hear the shape of a drum?

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Francesc Xavier Massaneda Clares i Joaquim Ortega Cerdà[en] In this work we study Mark Kac’s classical problem “Can one hear the shape of a drum?” and some of its extensions. They are all inverse problems on characterizing the shape, or at least some geometrical information about the shape, of an Euclidean domain from its Dirichlet spectrum. As to the original problem, we answer it negatively by providing an example of two different shaped planar drums that have the same spectrum of frequencies. As to the extensions, we prove that the spectrum of frequencies of a planar drum characterizes its area. These results are straightforwardly generalized to higher dimensions. Finally, we comment variants of Kac’s problem for which there are positive results for the characterization of the shape of a drum from its spectrum

Divergent perturbative series and Oppenheimer’s formula

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022-2023, Tutor: Bartomeu Fiol NúñnezWe present a derivation of the decay rate of a Hydrogen-type atom subjected to an external electrical field, the so-called Oppenheimer’s formula. The perturbative approach to this problem yields a divergent result, so we introduce the necessary mathematical tools to obtain a finite answer, and we illustrate them with two toy models. The result obtained agrees with the answer obtained originally by a WKB analysis