Spectral Simplicity and Asymptotic Separation of Variables

Abstract

We describe a method for comparing the real analytic eigenbranches of two families, (at) and (qt), of quadratic forms that degenerate as t tends to zero. We consider families (at) amenable to 'separation of variables' and show that if (qt) is asymptotic to (at) at first order as t tends to 0, then the generic spectral simplicity of (at) implies that the eigenbranches of (qt) are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional.L.H. would like to thank the Indiana University for its invitation and hospitality and the ANR programs 'Teichmüller' and 'Résonances et chaos quantiques' for their support. C.J. thanks the

Erratum to: Spectral Simplicity and Asymptotic Separation of Variables

“…Despite this gap, Proposition 9.1 is correct and, in order to rectify the situation, we provide here the correct estimates needed to derive Proposition 9.1. As a consequence, the statements of the main results of the article [HllJdg11] are correct.…”

“…The statement of Lemma A.3 in [HllJdg11] is false. 1 This lemma is used in the proof of Lemma 8.2 which is also incorrect and is used in the proof of Proposition 9.1.…”

“…More precisely, it relies on the fact that a quasimode of order t for a μ t at a non-critical energy cannot concentrate on the turning point (and thus must have some mass in the classically allowed region). In the exposition given in [HllJdg11], this non-concentration was hidden behind Lemmas 9.4 and 9.5. We will make it more transparent here by directly using the Langer-Cherry transform and the following estimate for solutions to the semiclassical Airy equation: 2 …”

“…where A ± are the linearly independent Airy functions that are defined in the appendix of [HllJdg11], and w is their Wronskian determinant. We use the Cauchy-Schwarz inequality and rescale the integrals with A ± by t − 2 3 .…”

“…Let W E denote the Langer-Cherry transform of w at energy E, and let φ E : [0, ∞) → R denote the associated change of variables (see §7 in [HllJdg11]). Let…”