Spectral Simplicity and Asymptotic Separation of Variables

We describe a method for comparing the spectra of two real-analytic families, ( a t ) and ( q t ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family ( a t ) is amenable to ‘separation of variables’ and that each eigenspace of a t is 1-dimensional for some t . We show that if ( q t ) is asymptotic to ( a t ) at first order as t → 0, then the eigenspaces of ( q t ) are also 1-dimensional for all but countably many t . As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.