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In fact, we applied a slightly different procedure. After step (i) we knew only the asymptotic of T (s,x) at Rex 1 and | s- sc | 1. The analytic continuation of T (s,x) as a function of the variable s to the area s< sc has a singularity in terms of the variable x at large x. Therefore, to calculate ρ (λ) at λ> λc, one can use 50. Then the asymptotics of ρ (λ) at λ→ λc +0 is defined by the position of the singularity with respect to the variable x, and by the behavior of T (-λ,x) near this singularity.
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In fact, we applied a slightly different procedure. After step (i) we knew only the asymptotic of T (s,x) at Rex 1 and | s- sc | 1. The analytic continuation of T (s,x) as a function of the variable s to the area s< sc has a singularity in terms of the variable x at large x. Therefore, to calculate ρ (λ) at λ> λc, one can use 50. Then the asymptotics of ρ (λ) at λ→ λc +0 is defined by the position of the singularity with respect to the variable x, and by the behavior of T (-λ,x) near this singularity.
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41149164092
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Alternatively, find eigenfunctions and eigenvalues, ψm (s,x) and μm (s), of the integral equation 27. Then find P̄ l (s) by using Eqs. 31 32 33. Here the nontrivial part of the propagator, describing its relaxation to a stationary value, corresponds to the discrete part of the spectrum of characteristic numbers of this equation.
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Alternatively, find eigenfunctions and eigenvalues, ψm (s,x) and μm (s), of the integral equation 27. Then find P̄ l (s) by using Eqs. 31 32 33. Here the nontrivial part of the propagator, describing its relaxation to a stationary value, corresponds to the discrete part of the spectrum of characteristic numbers of this equation.
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