Espectro do operador Laplaciano de Dirichlet em tubos deformados

Abstract

Let Ω be a deformed tube in R3 and −∆Ω D the Dirichlet Laplacian operator in Ω. In this work, we are going to study the spectrum σ(−∆Ω D) of the operator −∆Ω D. More precisely, we are going to analize how the geometrical characteris- tics of Ω can influence in the set σ(−∆Ω D). Firstly, we are going to show that, under certain conditions, the essential spectrum σess(−∆Ω D) of −∆Ω D is the same, independent if the tube is straight, curved or twisted. In regard to the discrete spectrum σdis(−∆Ω D) of −∆Ω D, we are going to show that if Ω is a curved tube, then σdis(−∆Ω D) is a non empty set. Furthermore, if Ω is a twisted tube, then σdis(−∆Ω D) is a empty set. In the case where Ω is lightly curved and twisted simultaneously, we are going to see that the discrete spectrum remains empty.