It is shown that the ‘‘Calogero’’ bound on the number of bound states in an attractive monotonous potential is not optimal for a strictly positive angular momentum ℓ and a new bound including an extra additive term is proposed. It is Nℓ(V)<(2/π)∫0∞√‖V(r)‖dr+1−√1 +(2/π)2ℓ(ℓ+1). From this new bound it is possible to obtain a bound on the total number of bound states for arbitrary angular momentum. The situation for −1/2≤ℓ<0 is investigated and a bound under the condition that r2V(r) has a single extremum is given. Consequences for zero angular momentum bound states in two dimensions are discussed.
REFERENCES
1.
2.
J.
Schwinger
, Proc. Natl. Acad. Sci. USA
47
, 122
(1961
)., Proc. Natl. Acad. Sci. U.S.A.
3.
4.
See references quoted in Ref. 2.
5.
E. Hille, Lectures on Ordinary Differential Equations (Addison–Wesley, Reading, MA, 1969).
6.
M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1982).
7.
E. Lieb, Proc. AMS Conf. Honolulu, edited by S. H. Gould (American Mathematical Society, Providence, 1979).
8.
9.
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© 1995 American Institute of Physics.
1995
American Institute of Physics
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