2012/07/18 by Guy Bouchitté, M. L. Mascarenhas, Luisa Mascarenhas +2
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Boundary value problem #Curvature #Dirichlet boundary condition #Geometry #Infinitesimal #Laplace operator #Mathematical analysis #Mathematical physics #Mathematics #Mixed boundary condition #Numerical methods in inverse problems #Operator (biology) #Physics #Robin boundary condition #Spectral Theory in Mathematical Physics #Symmetry (geometry) #math-ph #math.AP #math.MP
paper · pdf · doi:10.1063/1.4768462
arxiv created 2012/07/18 · openalex publication_date 2012/12/01 · arxiv updated 2015/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the Laplace operator in a thin three-dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case [G. Bouchitté, M. L. Mascarenhas, and L. Trabucho, “On the curvature and torsion effects in one dimensional waveguides,” COCV 13(4), 793–808 (2007)10.1051/cocv:2007042], we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rotation angle. In the symmetric case, the behavior of lower energy modes is shown to be ruled by a one-dimensional Sturm-Liouville problem involving an effective potential given in explicit form.