Learning Lagrange Mesh Method for 1D Schrödinger Eqtn.

[ognik]

Hi, I have been trying to find some articles that would cover the Lagrange mesh method applied to the 1-D Schrödinger eqtn. - using the Laguerre mesh. I want to develop some fortran programs, for example building Lagrange functions, the kinetic energy matrix elements ...

LMM is totally new to me, but after a few days of searching through some article databases (Scopus, IOPscience etc.) I haven't found anything that makes enough sense to me; (I did get the idea that Laguerre mesh's are perhaps not ideal for the 1-D case)

So I'd appreciate any links to articles, or anything else, that are perhaps more basic and could give me a 1st step...

 

[Ackbach]

I can't say that I know anything, really, about LMM. However, I have done lots of research, so there's my claim to helpfulness. This paper has a quick summary. I'm not at a university, so I'm afraid the arxiv is my best bet at giving you an actual pdf. Here's a paper using the method on the Dirac equation. Additionally, this paper, which compares the LMM with another method, seems to go into more implementation details. That might be helpful. This paper combines LMM with the Jost Function Method, something I used in my dissertation; it also goes into more detail concerning the LMM. This article seems rather seminal (Baye's name appears on nearly every paper connected with LMM - it's reasonable to suppose he had a lot to do with it).

Hope that gets you going!

 

[ognik]

Thanks Ackbach, A couple of useful papers I had not found through my university databases.

I can now (just about) follow LMM through various articles, but my understanding (not unlike many of the articles) seems to trickle to an inconclusive state when it comes to how to practically implement it, IE what are the algorithms I will need. This is also the first time I have done something like this, so appreciate any mentoring.

I found what must be one of the seminal articles, by (the ubiquitous) Baye and Heenen, 1986 - ref1. I will probably take most of my formula from there, as they also briefly visit the Laguerre mesh. Here's what I think so far (Feel free to criticise):

I will assume the potential V everywhere = 0, then the eqtn I want to solve is (from ref1) $ \hat{T}\phi = E\phi $, where $ \hat{T} = - \d{^2{}}{{x}^2} + \frac{\alpha(\alpha -2)}{4x^2} $, I do wonder where the usual $\frac{{\hbar}^{2}}{2m}$ term has gone, I know that we use $\hbar = 1$ for convenience sometimes, but what of the 2m?

Also I want the 1-D Schrödinger, but I think the term in $\alpha$ above is for polar/radial co-ords, so am I only interested in $ \hat{T} = - \d{^2{}}{{x}^2} $?

Anyway, principally I want to approximate the kinetic matrix T using LMM. I would also like to find the (approximated) eigenvalues.

Ref1 then states:

$ T_{ij} =\frac{{\alpha + 1}^{2}}{(4x_i)^2} + S_{ij}, i=j $
$ T_{ij} = {(-1)}^{(i-j)} \left[ \frac{1}{2}\left(\alpha+1{\left(x_i x_j\right)}^{-\frac{1}{2}}\right)
\left( {x_i}^{-1}+{x_j}^{-1} \right) + S_{ij} \right] , i \ne j$

where $ S_{ij}=\left(x_ix_j\right)^{\frac{1}{2}} \sum_{k \ne i,j} {x^{-1}_k }(x_k - x_i)^{-1}(x_k-x_j)^{-1} $

I am expecting a diagonal-ish square matrix - but how large? i & j must vary from 1 to N, what order is N useful enough - $10^6 $? or $ 10^{60} $?...

Then from here on I should apply the normal variational principle, assume trial solutions for $\psi$ and vary $\alpha$ to produce eigenvalues?