We study solutions to the time-dependent linear Schrödinger equation[Special characters omitted.] with [Special characters omitted.] where the potential V ( x ) grows superquadratically at infinity. In the subquadratic case, the solution operator [Special characters omitted.] is known to exhibit dispersive smoothing (that is, gain of [Special characters omitted.] derivative when averaged in time). We demonstrate failure of such an effect in the superquadratic case when measured locally in phase space. To this end, we devise a new calculus of pseudodifferential operators called the quasi-isotropic calculus and further generalize the wavefront sets of Hörmander and Nakamura. We use this to define microlocal dispersive smoothing and prove a theorem in the case n = 1 in which such smoothing fails at all z ∈ T *[Special characters omitted.] \0 for all time t. In higher dimensions, we prove a theorem about infinite speed of propagation of a certain class of these wavefront sets, and finally, we generalize the one-dimensional result and show the geometry of the Hamiltonian flow determines the exact nature of the irregularity.
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