A Laplace operator with boundary condi-tions singular at one point
(14 déc. 2020/Dec. 14, 2020) Seminar Spectral Geometry / Séminaire Spectral Geometry https://archimede.mat.ulaval.ca/agirouard/SpectralClouds/ Marco Marletta (Cardiff School of Mathematics, UK) A Laplace operator with boundary condi-tions singular at one point Abstract: In this talk I will present some work with Rozenblum from 2009. While ithas been known for more than half a century that the Laplace operator on asmooth, bounded domain may have essential spectrum if the boundary con-ditions are suitably chosen, typical choices involved non-local operators. Inthis talk I will show, with very elementary arguments, that even local bound-ary conditions, singular even just at a single point - can have a huge impacton the spectrum and eigenfunctions. The example we consider, first proposedby Berry and Dennis. still has empty essential spectrum and compact resol-vent. However Weyl’s law fails completely because the spectrum becomesunbounded below. The positive eigenvalues still obey Weyl asymptotics, toleading order; however the (absolute values of the) negative eigenvalues donot obey a power law distribution. I will also make some remarks and ask some questions about nodal do-mains, which were not addressed in our paper.
(14 déc. 2020/Dec. 14, 2020) Seminar Spectral Geometry / Séminaire Spectral Geometry https://archimede.mat.ulaval.ca/agirouard/SpectralClouds/ Marco Marletta (Cardiff School of Mathematics, UK) A Laplace operator with boundary condi-tions singular at one point Abstract: In this talk I will present some work with Rozenblum from 2009. While ithas been known for more than half a century that the Laplace operator on asmooth, bounded domain may have essential spectrum if the boundary con-ditions are suitably chosen, typical choices involved non-local operators. Inthis talk I will show, with very elementary arguments, that even local bound-ary conditions, singular even just at a single point - can have a huge impacton the spectrum and eigenfunctions. The example we consider, first proposedby Berry and Dennis. still has empty essential spectrum and compact resol-vent. However Weyl’s law fails completely because the spectrum becomesunbounded below. The positive eigenvalues still obey Weyl asymptotics, toleading order; however the (absolute values of the) negative eigenvalues donot obey a power law distribution. I will also make some remarks and ask some questions about nodal do-mains, which were not addressed in our paper.