Dirichlet and neumann spectrum

Abstract

AbstractIt is well known that the critical points of the Rayleigh quotient for the Dirichlet form relative to the square integral norm are eigenfunctions of the Laplace operator with Neumann boundary conditions and eigenvalue equal to the critical value.Clearly then, if we fix a critical point and consider the corresponding value and restrict the Rayleigh quotient to the space of eigenfunctions of the Laplace operator with that given eigenvalue, then the critical point remains a critical point of the restriction.In this paper we answer the converse question. We show that if a given number occurs as a critical value on the subspace of eigenfunctions with that same number as eigenvalue, then either it is a critical value for the quotient on the entire space or it is a critical value on the subspace of all functions satisfying Dirichlet boundary conditions. Moreover, a precise relation holds between the corresponding critical points.A similar relation holds for any elliptic operator in divergence form. The relation stems from a lattice identity induced by a certain indefinite quadratic form, involving certain subspaces.This theory has been successfully applied to calculating the cuspidal spectrum in a certain range for the Laplace‐Beltrami operator for certain finite volume hyperbolic surfaces, addressing a conjecture of Roelcke and Selberg, and recent conjectures of Sarnak; see [1].The numerical method used can be applied quite generally, typically to equations related to underlying symmetries where decompositions of the group give rise to lots of formal solutions.