Exact Recursion Relation for 2 × N Arrays of Dumbbells
It is shown that A(q, N), the number of ways of arranging q indistinguishable dumbbells on a 2 × N rectangular array of compartments, is exactly described by the recursion relation.For large val...
It is shown that A(q, N), the number of ways of arranging q indistinguishable dumbbells on a 2 × N rectangular array of compartments, is exactly described by the recursion relation.For large val...
Diagrammatic Perturbation Expansion for Ensembles of Random Matrices
A method for obtaining a perturbation expansion of various eigenvalue distributions corresponding to a certain class of perturbed ensembles of random matrices is given. The terms in the expansion can ...
A method for obtaining a perturbation expansion of various eigenvalue distributions corresponding to a certain class of perturbed ensembles of random matrices is given. The terms in the expansion can ...
Extension of the Riemann-Lebesgue Lemma
J. Math. Phys. 11, 3099 (1970); doi:10.1063/1.1665099
Issue Date: October 1970
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We show that, in the limit of large 
, integrals of the form
, integrals of the formare essentially given by ![[integral]](http://scitation.aip.org/stockgif3/int.gif)
R![[prime]](http://scitation.aip.org/stockgif3/prime-script.gif)
[f(x)/u(x)]dx where the region R![[prime]](http://scitation.aip.org/stockgif3/prime.gif)
is the union of all those subintervals in which |u| 
1. The corrections to this expression are of two kinds: terms O(1/) which depend on the details of averaging to remove logarithmic singularities in H() and terms O[(ln )/]. Some examples are given. If |u| 
1, the leading term in H vanishes and H() is bounded by (ln )/.
©1970 The American Institute of Physics
| History: | Received 23 May 1969; revised 20 March 1970 |
| Permalink: |
http://link.aip.org/link/?JMAPAQ/11/3099/1 |
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REFERENCES (5)
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- A good reference for all these matters is M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1965), esp. Chaps. 3 and 6.
- E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge U.P., Cambridge, 1952), expecially Secs. 3.64 and 9.41.
- The results of that analysis will be published elsewhere. They are contained in P. B. Kantor, “Scattering From A Composite System; High Energy Limit of the Closure Approximation,” Case Western Reserve University, Cleveland, Ohio, Preprint, 1970.
- To see this, simply apply the inequality Vfg
Vg sup |f|+Vf sup |g| to each subinterval in which u is monotonic, and use induction on n. - L. L. Foldy and J. D. Walecka,
Ann. Phys. (N.Y.) 54, 447 (1969) .
