Operators on leaves of the foliation generated by locally free action of $\mathbb R$

For a manifold $M$ with foliation $F$, we construct an inclusion
$$ \phi:C_0(M)|_L\to C_0(L)\times\prod_\mathbb ZC([0,1]) $$
where $L$ is a leaf of $F$ and $C_0(X)$ is the space of continuous functions with compact support. Using $\phi$, we study properties of operators on the spaces of functions on leaves of the foliation $F$. We also find properties of spectra of Schroedinger-type operators on the leaves of $F$.