# Asymptotic Analysis of 1-D Schrödinger Equation

+ 2 like - 0 dislike
169 views

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D:

$$y'' +V(x)y=Ey$$

$$y''=Q(x)y$$

where $Q(x):=E-V(x)$.

There is obviously nothing stopping me now from choosing arbitrary $Q(x)$ and finding solutions. However, I'm interested in coefficient functions that are actually relevant. I, being of weak physics background don't particularly know which $Q(x)$ would actually be both mathematically and physically intriguing. I'm most interested in evaluating the asymptotic behavior of solutions about some irregular singular point as well as one and two turning point problems. I then want to use Padé approximations and then potentially a Shanks transform on the asymptotic series associated with the asymptotic behavior. So my question stands, which $Q(x)$ should I choose? Many different recommendations would be much appreciated.

This post imported from StackExchange Physics at 2015-02-20 16:50 (UTC), posted by SE-user Seenathin
Have a look at the lectures by Carl Bender at pirsa.org; he discusses asymptotics, the Schrodinger equation, Pade approximations and the Shanks transform. He also addresses the issue of when $Q(x)$ is zero (which is a problem since the regular WKB approximation becomes singular).

This post imported from StackExchange Physics at 2015-02-20 16:50 (UTC), posted by SE-user JamalS

Typical potentials of interest are the quartic anharmonic oscillator, where $V(x)=ax^2+bx^3+cx^4$ and the Morse potential where $V(x)=a(1-e^{b(x-x_0)})^2$.
 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): Email me at this address if my answer is selected or commented on: 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:p$\hbar$ysicsOver$\varnothing$lowThen 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). To avoid this verification in future, please log in or register.