Method of steepest descent for a quartic oscillator

[Fabio Hernandez]

Homework Statement

Homework Equations

http://www.maths.manchester.ac.uk/~gajjar/MATH44011/notes/44011_note4.pdf

The Attempt at a Solution

I obtain i) and ii), but I can't understand the way to study iii) and iv), particularly to choose the rigth paths.[/B]

 

[DrDu]

in i and ii you have obtained a saddle point, but the directions in which the saddle will fall off (= steepest decent) will be different for the two cases lambda<0 and lambda >0. I.e. $F(z)\approx F(z_0)+F''(z_0)(z-z_0)^2/2$. If $F''(z_0)<0$, then the path of steepest descents will be for real z-z_0.