Modification of Effective Range Theory in the Presence of a Long Range 1r4 P
Modification of Effective Range Theory in the Presence of a Long Range 1r4 P
T F Omalley
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) B. Conditions for existence of r o If A(L) does not exist, the question of the existence of r (L) as defined by Eq. (1. 2) does not arise. If A(L) does exist, we must still examine the conditions under vhich r (L) will also exist. We will restrict ourselves to the o^ ' case L = and to n S ^. (it should not be very difficult to do the same for L > 0. ) To analyze the existence conditions, it will be useful to very briefly review effective range theoi'y for short range potentials. Of the deriva...tions 1 2 of the theory that have been given ', that of Bethe will be most convenient for our present purposes . Since we are concerned with L = 0, our boundary conditions, Eq. (2. 2) and {2. X), become u(0) = 0, u(r) ^ cot Ti(0) sin kr + cos kr (2. 2)' u (0) = 0, u (r) - 1 - r/A. (2.^)' o^ o We also introduce the solutions of the free Schroedinger equation, with wave numbers k and 0, to be denoted by v(r) and v (r), respectively, with normaliza- tions chosen so that they asymptotically approach u(r) and u (r), respectively.
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