Several methods of obtaining upper and lower bounds to the eigenvalues of self-adjoint operators bounded from below have recently been developed by Löwdin. All these procedures are based on a bracketing theorem. We mention them and discuss one of the variants that makes use of intermediate Hamiltonians. To illustrate the power of the method, an application is made to various forms of double-minimum (D.M.) potentials, both symmetric and asymmetric, which have already been studied by Somorjai and Hornig. The new results show the importance of obtaining these lower bounds in connection with the resonance interaction between accidentally coincident ``left'' and ``right'' levels of weakly asymmetric D.M. potentials. An agreement between upper and lower bounds of 12–15 significant figures is obtained.