This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term beyond all orders of a saddle point expansion (singular terms of an asymptotic series), like in the problem of the lifetime of a bound state in 1+0 negative coupling ϕ4 toy model.
Consider a particle with an initial (normalized) wave-function ψ0(x)=e−(x+e−x)/2 This specific shape defines a natural unit for x, note the double-exponential asymptotics of ψ0(x) as x→−∞.
Time evolution under the Hamiltonian H=−12∂2x transforms the wave-function to (using the textbook propagator)
ψ(x,t)=(2πit)−1/2∫ei(x−x′)2/(2t)ψ0(x′)dx′
My question is about the asymptotics of this integral, especially the leading front propagating to the left. Here is where I've hit the wall:
The saddle point expansion in t−1 gives t|ψ(x,t)|2∼e−e−x[1+(e3x−2e2x)t−1/8+O(t−2)] which converges nicely (checked numerically) for x≳1, but fails to capture the terms of order ex/t that dominate over the double exponential at negtavie x.
For t→+∞ the solution becomes symmetric, |ψ(x,t→∞)|2=1tcosh(πx/t)
Any ideas/hints will be appreciated.
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