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

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.

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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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Simon, Barry0000-0003-2561-8539
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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Air Force Office of Scientific Research (AFOSR)AF49(638)-1545
NSF Graduate Research FellowshipUNSPECIFIED
Issue or Number:8
Record Number:CaltechAUTHORS:20181119-161054026
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:91062
Deposited By: George Porter
Deposited On:20 Nov 2018 00:24
Last Modified:16 Nov 2021 03:38

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