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 .
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Communicated by B. Simon
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Zelditch, S. Reconstruction of singularities for solutions of Schrödinger's equation. Commun.Math. Phys. 90, 1–26 (1983). https://doi.org/10.1007/BF01209385
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DOI: https://doi.org/10.1007/BF01209385