Erratum to: “Spectral theory of Laplacians for Hecke groups with primitive character”

Abstract

In [1] we considered character perturbations of the automorphic Laplacian A=A(F0, X) for the Hecke group r0(N) with primitive character X. We assume that N=4N2 or N=4N3, where N2 and N3 are products of distinct primes and N2=2 mod 4, N3~3 rood 4. In these cases we are dealing with regular perturbations of A, which allows for a rigorous analysis of the problem of stability of embedded eigenvalues. The perturbation is represented on the form a M + a 2 N , where M is a first order differential operator and N is a multiplication operator. In order to prove instability of an embedded eigenvalue A we prove that the Phillips Sarnak integral I(O, A)={M~, E ) r for a common eigenfunction q~ of A with eigenvalue A and all Hecke operators, where E is a generalized eigenfunction with eigenvalue A. We consider only the operator Aodd acting on odd eigenfunctions, since (MeP, E}=0 for q) even. Let A = 8 8 2 be an eigenvalue of Aodd, and 0(q) the eigenvalues of the exceptional Hecke operators U(q), q IN, with the common eigenfunction ~. The operators U(q) are unitary ([1, Theorem 4.1]), so the eigenvalues 0(q) lie on the unit circle. The basic result on the Phillips Sarnak integral follows from [1, (7.23), (7.24)]. We formulate this in the following theorem.