### On the principal eigenvalue of elliptic operators in ${\mathbb{R}}^{N}$ and applications

Two generalizations of the notion of principal eigenvalue for elliptic operators in ${\mathbb{R}}^{N}$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semilinear problems in the whole space. We also indicate several outstanding open problems and formulate some conjectures.