Levinsons theorem

The values are taken from [7], where it is shown, that the correlated density contains the contribution of bound states as well the contribution of scattering states. Above the so called Mott density, where the bound states begin to disappear, according to the Levinson theorem, the continuous behavior of the correlated density is produced by the scattering states. 

One of these is Levinson s theorem (see section 5.14), which tells us that, for any short range well corresponding to an angular momentum ... 

However, we must take care that cot 8 is cyclic, whereas cothA is not. Thus, the more correct choice is 8 + mir = iA, when equation (5.16) is satisfied and the value of m must be chosen to satisfy Levinson s theorem and to allow a smooth continuation. 

This phase shift is a direct consequence of the higher frequency within the range of the potential. As the potential is made either wider or deeper, states are sucked in (fi-om the box) and are localized in the potential well. In the case of a one-dimensional square well of depth Vand width L, the number of bound states for a particle of mass misN= 1 + [ 2mV) L/(nh)] where the square brackets stand for the integer part. Every time the phase shift passes Jt, another one of the continuum (particle-in-a-box) states is sucked into the well. This phase shift, which plays a central role in scattering problems, records the asymptotic compression of the wave function, but the number of (1/2) oscillations (each producing a bound state) is lost. However, the number of bound states for each , N, can be recovered as it is encoded in the zero energy (i.e., threshold) phase shift, 5 (0) = N( n. This is known as Levinson s Theorem. 

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