Statistics of the half-integer stopband

We consider the statistical nature of the stopband set up in the vicinity of a half-integer tune when a linear lattice is subjected to quadrupole errors distributed according to gaussians. The probability density function of the stopband, treated as a complex number, is found to be a correlated bivariate gaussian in the real and imaginary parts. The mean magnitude of the stopband is calculated in terms of the complete elliptic integral of the first kind, and the conditional probability density of its magnitude is obtained in closed form. A number of limiting conditions are studied. Finally, we estimate the requirements on a correction system for neutralizing the stopband with a given probability of success.<
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