Abstract

Consider the higher order parabolic operator ${\partial }_t+{(-{\Delta }_x)}^m$ and the higher order Schrödinger operator $i^{-1}{\partial }_t+{(-{\Delta }_x)}^m$ in $X=\{(t,x)\in {\mathbb{R}}^{1+n}; |t|<A,|x_n|<B\},$ where m and n are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when $x_n>0,$ then the solutions vanish in X. Such results are global if n > 1, and we also prove some relevant local results.

中文翻译：

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高阶抛物线方程和薛定谔方程通过超平面的唯一延续

摘要

考虑高阶抛物线算子 ${\partial }_{吨}+{(-{\Delta }_X)}^{米}$ 和高阶薛定谔算子 ${\mathrm{一世}}^{-1}{\partial }_{吨}+{(-{\Delta }_X)}^{米}$ 在 $X=\{(吨,X)\in {\mathbb{电阻}}^{1+n}; |吨|<\mathrm{一种},|X_n|<乙\},$其中m和n是任何正整数。在某些低阶和正则性假设下，我们证明如果线性问题的解在$X_n>0,$那么解在X 中消失。如果n  > 1，这样的结果是全局的，我们也证明了一些相关的局部结果。