Erratum to: Spectral Simplicity and Asymptotic Separation of Variables

Abstract

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. 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.We would like to emphasize that the overall strategy of the proof of Proposition 9.1 remains unchanged. In particular, Proposition 9.1 is a statement about concentration properties of quasimodes for the quadratic form a μ t coming from separation of variables. 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 The online version of the original article can be found under

Hyperbolic triangles without embedded eigenvalues

“…In this section we apply the method of asymptotic separation of variables developed in [HlrJdg11] (see also [HlrJdg12]) to the family q t . Using the small t asymptotics derived in §5, we approximate q t to first order with a family of quadratic forms a t for which separation of variables apply.…”

“…If > 0 this follows from Proposition 9.1 of [HlrJdg11] with µ = (π ) 2 and σ(y) = y −2 . See the end of [HlrJdg12] for a proof of Proposition 9.1 of [HlrJdg11].…”