The Point Spectrum of Smooth Noncompact Hyperbolic Surfaces with Finite Area

TL;DR

This paper investigates the spectral properties of smooth noncompact hyperbolic surfaces with finite area, focusing on boundary value problems and the Dirichlet-to-Neumann operator, providing explicit formulas and computational methods.

Contribution

It introduces a sequence of boundary value problems on hyperbolic surfaces and derives explicit symbol expansions for the Dirichlet-to-Neumann operator, enabling efficient computation.

Findings

Stable sesquilinear forms at cusps

Explicit symbol expansion formulas for Dirichlet-to-Neumann operator

Computational methods for symbol calculation

Abstract

We construct a sequence of boundary value problems on compact subsets of smooth noncompact hyperbolic surfaces of finite area. We prove that the sesquilinear forms associated to these boundary value problems are stable as well as consistent at continuous functions which vanish at cusps. We also give an explicit form for the symbol expansion of the Dirichlet-to-Neumann operator of a certain Schrodinger operator. The symbols appearing in the expansion of this Dirichlet-to-Neumann operator can be calculated quickly by a computer using the formulas we provide in this paper.