• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,757 comments
1,470 users with positive rep
818 active unimported users
More ...

  Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

+ 3 like - 0 dislike

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.

This post has been migrated from (A51.SE)

asked Mar 6, 2012 in Theoretical Physics by Slaviks (610 points) [ revision history ]
retagged Mar 25, 2014 by dimension10
Minor typo in the question(v3): The square root $(2\pi it)^{1/2}$ in the second formula should be in the denominator.

This post has been migrated from (A51.SE)
@Qmechanic Thanks, fixed this one! I'm prone to typos :(

This post has been migrated from (A51.SE)

1 Answer

+ 3 like - 0 dislike

After some struggle and a useful hint form a colleague, the problem finally cracked:

  1. Going to momentum space gives $$\psi_{0k} = \int \psi_0(x) e^{i k x} dx= 2^{-ik+1/2} \Gamma(-ik+1/2)$$

  2. Applying time evolution with $\mathcal{H}=k^2/2$ gives $$\psi(x,t) = \frac{1}{2\pi} \int e^{-i k^2 t/2-i k x} \psi_{0k} dk$$

  3. Using large-$z$ asymptotic expansion for $\Gamma(z+1)$ and identifying a stationary phase point near $k \approx -x/t$ give the leading order which coincides with the asymptotics of $t \gg 1$ solution: $$\psi(x,t) \sim (2/t) e^{\pi x/t} $$

  4. Finally, the prefactor is recovered by keeping all leading logs in the expansion of $\Gamma(z+1)$ and solving for the stationary point using Lambert W function (for which Mathematica nicely handles the asymptotics), gives the final answer $$\psi(x,t) = (2/t) e^{\pi x/t} (-2 x/t)^{\pi/t} + O(t^{1+\epsilon})$$ with $\epsilon>0$, valid for $x \ll -t$ both for $t$ large and small.

Thanks to all who paid attention and helped with advice.

This post has been migrated from (A51.SE)
answered Mar 8, 2012 by Slaviks (610 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights