Unique continuation through hyperplane for higher order parabolic and Schr\"odinger equations

Consider the higher order parabolic operator ${\mathrm{\partial }}_t+(-{\mathrm{\Delta }}_x{)}^m$ and the higher order Schr"{o}dinger operator $i^{-1}{\mathrm{\partial }}_t+(-{\mathrm{\Delta }}_x{)}^m$ in $X=\{(t,x)\in {\mathbb{R}}^{1+n};\mathrm{\mid }t\mathrm{\mid }<A,\mathrm{\mid }{x}_n\mathrm{\mid }<B\}$, 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.