ϱBootstrap in the Unitarized Strip Approximation

We present a method of improving the new strip approximation of Chew and Jones by calculating parts of the elastic double spectral functions using the Mandelstam iteration procedure. These double spectral functions are used to obtain additional contributions to the left-hand cuts of the partial-wave amplitudes, and also to estimate the inelasticity within the strip region. The inelastic $\frac{N}{D}$ equations are solved in the way proposed by Frye and Warnock. The method is applied to the problem of bootstrapping the $\rho$ trajectory in $\pi -\pi$ scattering, and some preliminary results are presented. We find that it is possible to obtain a self-consistent trajectory with the correct physical mass, width, and intercept $\alpha \mathrm{}\left(0\right)\mathrm{}$, but that the solution is by no means unique, since self-consistency can be achieved with trajectories having intercepts anywhere from $\alpha \mathrm{}\left(0\right)=1\mathrm{}$ to $\alpha \mathrm{}\left(0\right)\mathrm{}\approx 0.2$. Also, the trajectories have a large curvature, and large residue, which result in a violation of crossed-channel unitarity for low $l$.