Extension of the notion of a gap to differential operators defined on different open sets

In this paper the notion of a gap between two linear operators is extended to the case of linear differential operators defined on different open sets. Estimates of the gap between second order uniformly elliptic partial differential operators subject to homogeneous Dirichlet boundary conditions defined on different open sets $\Omega_1$ and $\Omega_2$ are obtained in terms of the geometrical characteristics of vicinity of $\Omega_1$ and $\Omega_2$. These estimates can be used for obtaining spectral stability estimates for the eigenvalues and eigenfunctions of the aforementioned operators.