On the spectrum of the Neumann Laplacian of long-range horns: a note on the Davies-Simon theorem

Abstract:

For a class of regions with cusps (e.g., $\Omega = \{ (x,y):x > 1,|y| < \exp ( - {x^\alpha })\} ,\;0 < \alpha < 1)$) we show that ${\sigma _{\operatorname {ac} }}( - \Delta _N^\Omega ) = [0,\infty )$ of uniform multiplicity one, ${\sigma _{\operatorname {sing} }}( - \Delta _N^\Omega ) = \emptyset$, and ${\sigma _{\operatorname {pp} }}( - \Delta _N^\Omega )$ consists of a discrete set of embedded eigenvalues of finite multiplicity.