Reconstruction of singularities for solutions of Schrödinger's equation
Tóm tắt
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
.
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