Sum Rules in the Dispersion Theory of Nuclear Reactions

In the first five sections the dispersion theory is developed with an internal region $V$ whose boundary $S$ is quite close to the nuclear surface. Two types of quantities then occur: those like the derivative matrix $R$, which connect the values and derivatives of the wave function at $S$, and those like the collision matrix $U$ which give the asymptotic behavior of the wave function. These latter are, of course, independent of the position of $S$. Because of the proximity of $S$ to the nuclear surface, the wave function in the closed channels and its derivative remain appreciable at $S$. They may, however, be eliminated from the formalism, and this is done in Sec. II, leading to a reduced $R$ which connects the values and derivatives of the wave function in the open channels only. In the next section this reduced $R$ is used to obtain expressions for the $S$-independent quantities, in particular, for the collision matrix. These expressions are given more explicit, but approximate form in Sec. IV. The development shows that the quantity, which in the usual formulas for the cross sections is interpreted as the nuclear radius, need not be equal to this at all, but is the distance at which two opposite effects compensate. The fifth section gives exact expressions for the poles and residues of some $S$-independent quantities, which are then compared with the poles and residues of the approximate expressions of the previous chapter. In this way criteria are derived for the accuracy of these approximate expressions.

The last sections contain derivations of two sum rules for the parameters in the original $R$. The consequences of these sum rules are traced for the reduced $R$ and for the $S$-independent quantities, in the light of the development of the first five sections. The first sum rule gives a maximum for the partial widths of levels, while the second leads to the well-known proportionality of reduced level width and level spacing. The meaning and validity of this is discussed in some detail, both for the single particle and many particle pictures.