MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$z(s)=1+\gamma\int\limits_{-\infty}^s ds_1\int\limits_{-\infty}^{s_1}ds_2 \cos{(s_1^2-s_2^2)}z(s_2)\;?$$ In fact I need to show that $$\lim_{s\to\infty} z(s)=2\exp{\left(\frac{\pi\gamma}{4}\right)}-1.$$ The integral equation is equivalent to the following third order differential equation $$sz^{\prime\prime\prime}-z^{\prime\prime}-s(\gamma-4s^2)z^{\prime}+ \gamma z=0,$$ and the initial conditions $z(-\infty)=1,\,z^\prime(-\infty)=0,\,z^{\prime\prime}(-\infty)=\gamma$.

The question arose in the context of remarkable connection between the Landau-Zener problem and the ball rolling along the Cornu spiral (that's how I do now what the $\lim_{s\to\infty} z(s)$ should be) established by Bloch and Rojo in

share|cite|improve this question
The paper cited by OP can be freely downloaded at – Jon Sep 24 '13 at 12:59
Try substituting $ z = \exp( f(s) ) $ in the differential equation and then look for terms that balance as $ s \rightarrow \infty $. This often allows for an asymptotic solution to be found. – Tom Dickens Sep 25 '13 at 3:28

The idea is to iterate on the integral equation starting with $z(s)=1$. This will give the integrals in a series $$ I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty }^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { (s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}. $$ The paper discussing them is this one. The general formula is $$ I_n=\frac{2}{n!}\left(\frac{\pi}{4}\right)^n $$ and you should take into account also powers of $\gamma$. These are the terms of the power series of the exponential multiplied by a factor 2.

share|cite|improve this answer
My goal was to calculate $I_n$ in a different way (without using the known Landau-Zener formula) than discussed in the paper. I hope some method exists to get the asymptotic behaviour of $z(s)$ from the integral equation quoted (or from the equivalent differential equation). Then this will give $I_n$ and therefore another derivation of the Landau-Zener formula, which was the final goal. – Zurab Silagadze Sep 24 '13 at 15:48

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.