Levinson’s theorem

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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