Bootstrap Conditions in a Soluble Model

A soluble model obtained by a slight extension of the Lee model is considered in a study of the bootstrap mechanism. By examining the general solution that is obtained by using properties of the Herglotz function, it is found that the bootstrap mechanism can be achieved if and only if two further restrictions in addition to the general requirements of analyticity, unitarity, and crossing symmetry are imposed on the solution. They are that (i) the scattering amplitude satisfy the asymptotic condition $limit\; of{\omega }^{-2\mathrm{}}{t}^{-1\mathrm{}}\left(\omega \right)\text{as}\omega \to \infty =0\mathrm{}$ and (ii) the scattering amplitude have no Castillejo-Dalitz-Dyson (C.D.D.) zeros. It is also proved that the condition (i) is equivalent to $limit\; of{\omega }^{-1\mathrm{}}D\left(\omega \right)\text{as}\omega \to \infty =0\mathrm{}$ when $N\left(\omega \right)=O\left({\omega }^{-1\mathrm{}}\right)$ as $\omega \to \infty$ or $Z_3=0\mathrm{}$ or the Levinson theorem holds, while condition (ii) is equivalent to assuming the two familiar bootstrap equations based on the $\frac{N}{D}$ method and implies in particular a nonpositive scattering length. Either of the conditions (i) and (ii) alone gives in general only an inequality between the mass and coupling constant, and it is therefore concluded that the possibility of the bootstrap mechanism depends in a very sensitive way on the low-energy behavior as well as the high-energy behavior of the scattering amplitude. It is further argued that destroying the crossing symmetry in the approximate solutions will not give any physically meaningful conditions for determining the parameters unless one introduces a subtraction or one C.D.D. zero in the Low equation.