Conditions for the existence of solutions of the equations of the type of the differential method equations

G. N. Chermalykh

Teoretičeskaâ i matematičeskaâ fizika, Tome 10 (1972) no. 2, pp. 215-222

Résumé

Equations of the type of the differential method equations are regarded as singular linear integral equations for the inelasticity coefficients under the assumption that the real parts of the phase shifts are known. For any finite system the existence of a unique solution (except for the CDD polynomial ambiguity) is proved under fairly weak restrictions on the dependence of the elements of the crossing matrix, $\beta_{ll'}(\omega,\omega')$ on $\omega$ and $\omega'$ ($l$, and $l'$ are the angular momenta and $\omega$ and $\omega'$ are the cms energies of the direct and crossed channels, respectively). Iris also shown that the condition that there exist a solution (in the class of continuous bounded functions) of the linear integral equation equivalent to an infinite system leads to the restriction $\beta_{ll'}(\omega,\omega')\to0$, $ll'\to\infty$, $\omega$, $\omega'\in[\omega_i,\infty)$, where $\omega_i$ is the inelastic threshold and to a behavior of the partial amplitude $T_l(\omega)\xrightarrow[l\to\infty]{}0$ $(\omega\in[\omega_i,\infty))$, characlteristic for strong interactions and which is usually deduced from the axiomatic $s\bigotimes t$ analyticity. Models with a short-range repulsion are discussed and allowance is made for inelasticity in the models of the differential method.