  We consider a second order self-adjoint operator in a domain which can be
bounded or unbounded. The boundary is partitioned into two parts with Dirichlet
boundary condition on one of them, and Neumann condition on the other. We
assume that the potential part of this operator is non-negative. We add a
localized perturbation assuming that it produces two negative isolated
eigenvalues being the two lowest spectral values of the resulting perturbed
operator. The main result is a lower bound on the gap between these two
eigenvalues. It is given explicitly in terms of the geometric properties of the
domain and the coefficients of the perturbed operator. We apply this estimate
to several asymptotic regimes studying its dependence on various parameters. We
discuss specific examples of operators to which the bounds can be applied.
