Details

Autor(en) / Beteiligte

• Wunsch, Jared

Titel

Microlocal analysis of the time-dependent Schroedinger operator

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

1998

Link zum Volltext

• ProQuest

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• We study the time-dependent Schrodinger equation$$\left({1\over i}\ {\partial\over \partial t} + {1\over 2}\Delta + V\right) \psi = 0$$for the (non-negative) Laplacian of a scattering metric on a compact manifold with boundary. Under a non-trapping hypothesis on the geodesics, the microlocal smoothness of $\psi(t,\cdot)$ for $t > 0$ is determined by the smoothness and growth properties of $\psi(0,\cdot)$ as measured by the "quadratic-scattering" wavefront set, a generalization of Hormander's wavefront set. We prove a propagation theorem for the quadratic-scattering wavefront set which describes singularities and growth of $\psi(t,\cdot)$ in terms of singularities and growth of $\psi(0,\cdot).$ We work with two classes of potentials V: perturbations of the free particle and perturbations of the harmonic oscillator. In the latter case, the propagation of singularities implies a trace theorem for the solution operator $e\sp{\rm -it({1\over 2}\Delta + V)}.$