Simon, Barry (1969) On the Growth of the Ground‐State Binding Energy with Increase in Potential Strength. Journal of Mathematical Physics, 10 (8). pp. 1415-1421. ISSN 0022-2488. doi:10.1063/1.1664983. https://resolver.caltech.edu/CaltechAUTHORS:20181119-161054026
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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 → ∞.
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Additional Information: | © 1969 American Institute of Physics. (Received 7 February 1969) The author would like to thank M. Reed for an enlightening conversation on the nonpower growth of convex functions. This research partially sponsored under Air Force Office of Scientific Research under Contract AF49(638)-1545. | ||||||
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Issue or Number: | 8 | ||||||
DOI: | 10.1063/1.1664983 | ||||||
Record Number: | CaltechAUTHORS:20181119-161054026 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20181119-161054026 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 91062 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | George Porter | ||||||
Deposited On: | 20 Nov 2018 00:24 | ||||||
Last Modified: | 16 Nov 2021 03:38 |
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