Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant

Abstract

The two lowest eigenvalues E 0( λ), E 1( λ) of a symmetric double well tunnelling problem − Δ + λ 2 V as λ → ∞ are considered and they are compared to the two lowest eigenvalues E ̃ 0(λ), E ̃ 1(λ) of −Δ + λ 2(V + W), where W is supported away from the well-bottoms of V. We determine the leading exponential splitting of various differences of the four numbers E 0, E 1, E ̃ 0, E ̃ 1 . Related problems are discussed.