Reconstruction of singularities for solutions of Schrödinger's equation

Abstract

We determine the behavior in time of singularities of solutions to some Schrödinger equations onR ⁿ. We assume the Hamiltonians are of the formH ₀+V, where$H_0 = 1/2\Delta + 1/2 \sum\limits_{k = 1}^n { \omega _k^2 x_k^2 }$, and whereV is bounded and smooth with decaying derivatives. When all ω _k =0, the kernelk(t,x,y) of exp (−itH) is smooth inx for every fixed (t,y). When all ω₁ are equal but non-zero, the initial singularity “reconstructs” at times$t = \frac{{m\pi }}{{\omega _1 }}$ and positionsx=(−1)^m y, just as ifV=0;k is otherwise regular. In the general case, the singular support is shown to be contained in the union of the hyperplanes$\{ x|x_{js} = ( - 1)^l js_{y_{js} } \}$, when ω _j t/π=l _j forj=j ₁,...,j _r .