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 $\phi^4$ toy model.

Consider a particle with an initial (normalized) wave-function $$\psi_0(x) = e^{-(x+e^{-x})/2}$$ This specific shape defines a natural unit for $x$, note the double-exponential asymptotics of $\psi_0(x)$ as $ x \to -\infty$.

Time evolution under the Hamiltonian $\mathcal{H}=-\frac{1}{2}\partial_x^2 $ transforms the wave-function to (using the textbook propagator)

$$\psi(x,t) = (2 \pi i t)^{-1/2} \int e^{i (x-x')^2/(2t)} \psi_0(x') d x'$$

**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 $$t |\psi(x,t)|^2 \sim e^{-e^{-x}} \left [1 + (e^{3x} -2 e^{2x}) \, t^{-1} /8 + O(t^{-2}) \right ] $$ which converges nicely (checked numerically) for $x \gtrsim 1$, but fails to capture the terms of order $e^{x/t}$ that dominate over the double exponential at negtavie $x$.

For $t \to +\infty$ the solution becomes symmetric, $$|\psi(x,t \to \infty)|^2=\frac{1}{t \cosh (\pi x/t)}$$

Any ideas/hints will be appreciated.

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