“…Recently, an important breakthrough has been made in understanding the essential spectrum of L and e tL , see [FV,FV2,FSV,FSV2,LH1,LH2,V,VF] and the bibliography therein. In particular, using asymptotic expansions for integral Fourier operators, the boundary of the essential spectrum of e tL (in dimensions two and three) was related to the maximal Lyapunov exponent of a so-called bicharacteristic amplitude system, see [V,VF,S2] and also (8) below.…”