Details

Autor(en) / Beteiligte

• Qian, Randy Zhigang

Titel

Diffractive theorems for the wave equation with inverse square potentials

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

2009

Link zum Volltext

• ProQuest

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• This dissertation investigates the phenomenon of diffraction resulting from the addition of an inverse square potential term to the wave operator. In particular, it explicitly establishes the existence of diffraction for the solution to the wave operator with an inverse square potential in 2-dimensional euclidean space and proves a propagation of smoothness result in two more general settings. Chapter 2 establishes diffraction in the fundamental solutions to the wave operator plus inverse square potential with a Dirac Delta initial condition in 2-dimensional euclidean space. Following methods as described by Cheeger and Taylor, we separate variables, apply spectral transforms to each variable, and employ contour deformation techniques to establish an explicit form for diffractive front in the fundamental solution. Chapter 3 proves a propagation of smoothness result for a related wave operator with potential, where instead of a constant, we put a smooth bounded function in the numerator of the potential. Microlocal energy estimates are used following the basic propagation methods of Duistermaat and Hörmander, and employing the heavy refinements due to Melrose, Vasy, and Wunsch to handle propagation through the radial point at the origin. The potential term is estimated using Hardy's Inequality. Chapter 4 extends the propagation of smoothness result to conic manifolds with an inverse square potential concentrated at their boundary. We state a product decomposition theorem for the conic metric due to Melrose and Wunsch, then use the resulting coordinates to deploy our argument from Chapter 3. New terms with dependence on distance to the boundary arise, and we show how to bound them.