 
Summary: CONVERGENCE OF AN EXACT QUANTIZATION SCHEME
ARTUR AVILA
Abstract. It has been shown by Voros [V1] that the spectrum of the onedimensional homo
geneous anharmonic oscillator (Schr¨odinger operator with potential q2M , M > 1) is a fixed
point of an explicit nonlinear transformation. We show that this fixed point is globally and
exponentially attractive in spaces of properly normalized sequences.
1. Introduction
Let 0 < < be a constant. For E, E > 0, define
(1.1) (E , E) = tan1 sin
E E1 + cos
.
Let X = (Xk)
k=1, Y = (Yj )
j=1 be sequences of positive real numbers and define = (j )
j=1
by
(1.2) j(X, Y ) =
1
k
