The Lagrange Mesh Technique
by Arjun Jain
As part of my final year project, I am studying about the Lagrange Mesh Technique.
The Lagrange-mesh technique is an approximation method, adapting Gaussian Quadrature to solve differential equations, particularly quantum-mechanical bound-state and scattering problems. There is no need for analytically evaluating matrix elements and the accuracy is exponential in the number of mesh points.
1. Gaussian Quadrature:
- Gaussian Quadrature is the starting point of the Lagrange mesh technique. Here, integrals whose integrands are known only at a few points of the domain are estimated by approximating the integrands by the Lagrange interpolating polynomials, defined as: . Note that . For a function , the approximation used is . Obviously, .
- Consider the integral . Here is a weight function, positive in the domain of integration. Substituting the expression of , we get , where , also known as the Christoffel symbols.
- The n-point quadrature is exact for polynomials upto degree , but Gauss extended the method to give exact results for degree . This is done by introducing restrictions on the points where the function is to be known. For this, a family of orthogonal polynomials is chosen, orthogonal with respect to the inner product: . For an n-point Gaussian quadrature then, the s are chosen as the zeroes of .
- As I prove in my project by induction, any polynomial orthogonal to all polynomials upto order in a given interval with respect to a weight function must have at least roots in , as is required for Gaussian Quadrature to work.
2. Lagrange functions:
- The Lagrange mesh technique uses the idea of Gaussian Quadrature to solve differential equations. For this, a set of Lagrange functions are defined over associated with mesh points over this interval. To the mesh is associated a Gaussian Quadrature.
- The Lagrange functions have the following properties:
1. ( is a normalizing wieght defined below).
2. Gaussian Quadrature is exact for products .
Property 2 implies that .
- If the corresponding Gaussian Quadrature involves the family of orthogonal polynomials with weight function , then explicitly , where , where is the Christoffel symbol corresponding to .
3. Schrödinger’s Equation, using the Lagrange Mesh Technique:
- Consider Schrödinger’s eigenvalue equation in : and approximate by .
- Using Gaussian quadrature to estimate the integrals got by multiplying the above by and integrating, we get , where is the potential energy matrix element and , the kinetic energy matrix element.
- Note that the potential matrix is diagonal, where each diagonal element need only be evaluated at a single mesh point and the kinetic matrix must be calculated only once for a given Lagrange basis, thus reducing computation time significantly.
Some examples of Lagrange meshes are meshes obtained using classical orthogonal polynomials like Legendre, Hermite, Laguerre. We can also use non-classical orthogonal polynomials to define other meshes like the Fourier mesh, sinc mesh, and the shifted Gaussians mesh. For the multidimensional case, we can also define a multidimensional Lagrange function as a product of 1-D Lagrange functions: .
1. Brandon Charles Bryant, Lagrange Meshes in Hadronic Physics, Honors in the Major Program, Theses, Paper 6, Florida State University, 2011
2. Daniel Baye, Lagrange-mesh method for quantum-mechanical problems, physica status solidi (b) 243.5 : 1095-1109, 2006
3. Daniel Baye and M. Vincke, Lagrange meshes from nonclassical orthogonal polynomials, Physical Review E, Volume 59, Number 6, June, 1999