  The classical Szeg\H{o}-Verblunsky theorem relates integrability of the
logarithm of the absolutely continuous part of a probability measure on the
circle to square summability of the sequence of recurrence coefficients for the
orthogonal polynomials determined by the measure. The present paper constructs
orthogonal polynomials on the torus of arbitrary finite dimension in order to
prove theorems of Szeg\H{o}-Verblunsky type in the multivariate and almost
periodic settings. The results are applied to the one-dimensional Schr\"odinger
equation in impedance form to yield a new trace formula valid for piecewise
constant impedance, a case where the classical trace formula breaks down. As a
byproduct, the analysis gives an explicit formula for the Taylor coefficients
of a bounded holomorphic function on the open disk in terms of its continued
fraction expansion.
