On the Growth of the Ground‐State Binding Energy with Increase in Potential Strength

Abstract

We study the asymptotic behavior of the ground‐state binding energy G(λ) of −Δ + λV as λ → ∞. Unlike the number of bound states, G(λ) does not have a universal power growth as λ → ∞. It is shown, however, that as λ → ∞ for Kato potentials Aλ<G(λ)<Bλ^4 . Examples are presented for which G ∼ λ^β for any 1 < β < 4. Other examples are presented which obey no power growth. We also prove theorems which reflect the close connection between the large λ behavior of G and the small r behavior of V for potentials with a single attractive singularity at r = 0. These can be roughly phrased as follows: If V ∼ −r^(−α) for r → 0, then G(λ) ∼ λ^β with β = 2/(2 − α) as λ → ∞.