<A HREF="https://www.projecteuclid.org/journals/communications-in-mathematical-physics/volume-75/issue-3/Double-wells/cmp/1103908148.full">Double wells,</A> Evans M. Harrell, Comm. Math. Phys. 75, 239 (1980), should be sufficiently rigorous. The result, theorem 2.22, applies to a symmetric double-well potential in arbitrary number $n$ of dimensions, constructed as the sum of two single-well potentials, bounded and of compact support, after translation by $\pm f$.