In this work we consider an example of a linear time degenerate Schrödinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates as the linear solutions.

中文翻译：

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时简薛定谔算子的平滑效果

在这项工作中，我们考虑一个线性时间退化薛定谔算子的例子。我们表明，在适当的假设下，算子满足 Kato 平滑效果。我们还表明，涉及该算子和多项式导数非线性的非线性初值问题的解是局部适定的，并且它们的解也满足与线性解相同的平滑估计。