A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $ℒu=Vu$ where $V\in C^{\infty }$ and $ℒ={\sum }_{j=1}^{N-1}{X}_j^2$ is a (nonelliptic) operator in ${\mathbf{R}}^N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $ℒ$ is the subelliptic Laplacian on the Heisenberg group and $V$ is a zero order term which is allowed to be unbounded. We provide a sufficient condition, involving a first order differential inequality, for nontrivial solutions of $ℒu=Vu$ to have a finite order...