MHB Equations for Lagrange-Laguerre mesh

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The discussion focuses on the confusion surrounding the practical equations for the Kinetic energy matrix elements using the Laguerre mesh in solving the 1-D Schrödinger equation. The user seeks clarification on the appropriate form of the kinetic operator, particularly whether to use the term $\hat{T} = -\frac{d^2}{dx^2}$ without the additional radial term. There is uncertainty about the size of the matrix, with the user questioning how to determine the value of N, which influences both the matrix dimensions and the accuracy of the method. The user notes that the choice of N affects the eigenvalues and eigenvectors, leading to concerns about potential errors due to ignoring terms between N and infinity. Overall, the thread highlights the complexities of implementing the Laguerre mesh method in a 1-D context.
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Hi - I have been through quite a few articles on the Lagrange Mesh method and mostly follow it, but still find it confusing to understand which are the practical equations I should be using.

I want to find the expression(s) for the Kinetic energy matrix elements for the Laguerre mesh to be used in solving the 1-D Schrodinger eqtn. - so that I can write a fortran program to calculate the elements.

I will assume the potential V everywhere = 0, then the Schrodinger eqtn I want to solve is (from a couple of the articles, mostly with Baye an author) $ \hat{T}\phi = E\phi $, where $ \hat{T} = - \d{^2{}}{{x}^2} + \frac{\alpha(\alpha -2)}{4x^2} $

Not stated in the article, but I will assume that the usual $\frac{{\hbar}^{2}}{2m}$ term has been set to 1 for convenience.

The articles actually recommend the Hermite mesh for 1-D, but my task is to use the Laguerre mesh. What I couldn't be certain of from the articles therefore, is that the $\frac{\alpha(\alpha -2)}{4x^2}$ term seems to be for the Laguerre mesh in a radial situation, therefore for the 1-D case I think I should use $\hat{T} = - \d{^2{}}{{x}^2}$ but would appreciate confirmation?Either way, the articles state the following eqtns for the matrix elements:

$ 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} $

But how big do I make the matrix? I think that I should sum i,j from 1 to N and look at the accuracy with different values of N
The Laguerre mesh is over $[0, \infty)$ and some of the error in the method are the mesh points between N and $\infty$ that we ignore. An example in an article used N=4, but with no justification.

However the column vectors of T will be an orthonormal basis for the space, which should be 1-D so I can't see how T can be an NxN matrix, with N large?

The N I choose also determines the Laguerre eqtn from which the mesh points $x_i$ can be determined, the roots from $L^{\alpha}_N(x_i) = 0 $ This all makes me think there is more than 1 N in play?

I have read through these articles until the letters dribbled off the page in protest, the answer to N eludes me.

Even if you don't have all the answers, I'd appreciate all assistance, thanks.
 
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I noticed that N determines the number of solutions (eigenvalues, leading to eigenvectors). I think that by choosing N, I effectively ignore the terms between N and $\infty$ - so those represent some error in the method.
I am still not sure what N represents, I could see it representing the radial distance from the original but am sure it is more than that?
 
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