We prove the existence of a complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrödinger operator acting in when the potential is real and either smooth periodic, or generic quasiperiodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.
"Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrödinger operators." Duke Math. J. 165 (3) 509 - 561, 15 February 2016. https://doi.org/10.1215/00127094-3166415