The Lagrange Mesh Technique

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.

• 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: $\displaystyle{\pi(x)=\prod_{j=1}^m(x-x_j)}$. Note that $\displaystyle{\pi'(x_j)=\prod_{i=1, i\neq j}^m(x_j-x_i)}$. For a function $f(x)$, the approximation used is $\displaystyle{\Phi(x)=\sum_{j=1}^m \frac{\pi(x)}{(x-x_j)\pi' (x_j)} f(x_j)}$. Obviously, $\Phi(x_j)=f(x_j)$.
• Consider the integral $\displaystyle{\int_a^b dx~ W(x) f(x) \approx\int_a^b dx~ W(x) \Phi(x) }$. Here $W(x)$ is a weight function, positive in the domain of integration. Substituting the expression of $\Phi(x)$, we get $\displaystyle{\int_a^b dx~ \left( W(x) \sum_{j=1}^m \frac{\pi(x)}{(x-x_j)\pi' (x_j)} f(x_j)\right) = \sum_{j=1}^m w_j f(x_j)}$, where $\displaystyle{w_j=\int_a^b dx~ W(x)\frac{\pi(x)}{(x-x_j)\pi' (x_j)}}$, also known as the Christoffel symbols.
• The n-point quadrature is exact for polynomials upto degree $n-1$, but Gauss extended the method to give exact results for degree $\leq 2n-1$. This is done by introducing restrictions on the points where the function is to be known. For this, a family of orthogonal polynomials $\{p_k\}$ is chosen, orthogonal with respect to the inner product: $\displaystyle{\left< f,g\right> = \int_a^b dx~ f(x)g(x)W(x)}$. For an n-point Gaussian quadrature then, the $x_j$s are chosen as the zeroes of $p_n(x)$.
• As I prove in my project by induction, any polynomial $p(x)$ orthogonal to all polynomials upto order $n-1$ in a given interval $(a,b)$ with respect to a weight function $W$ must have at least $n$ roots in $(a,b)$, as is required for Gaussian Quadrature to work.

2. Lagrange functions: $\newline$

• The Lagrange mesh technique uses the idea of Gaussian Quadrature to solve differential equations. For this, a set of $N$ Lagrange functions $f_i(x)$ are defined over $(a,b)$ associated with $N$ mesh points $x_i$ over this interval. To the mesh is associated a Gaussian Quadrature.
• The Lagrange functions have the following properties:

1. $f_i(x_j)=\lambda_i ^{-1/2} \delta_{ij}$ ( $\lambda_i$ is a normalizing wieght defined below).

2. Gaussian Quadrature is exact for products $f_i(x)f_j(x)$.

Property 2 implies that $\displaystyle{\int_a^b dx~ f_i(x)f_j(x) = \sum_{k=1}^N \lambda_k f_i(x_k)f_j(x_k) =\delta_{ij}}$.

• If the corresponding Gaussian Quadrature involves the family of orthogonal polynomials $\{ \Phi(x) \}$ with weight function $W(x)$, then explicitly $\displaystyle{f_i(x)=\frac{\lambda_i(x)^{-1/2} \Phi_N(x)}{\Phi_N'(x_i)(x-x_i)}}$, where $\displaystyle{\lambda_i(x)=\frac{w_i}{W(x)}}$, where $w_i$ is the Christoffel symbol corresponding to $x_i$.

3.  Schrödinger’s Equation, using the Lagrange Mesh Technique: $\newline$

• Consider Schrödinger’s eigenvalue equation in $1D$: $\displaystyle{\frac{-\hbar^2}{2m}(\frac{\partial^2\psi}{\partial x^2}) + V(x)\psi = E\psi }$ and approximate $\psi$ by $\displaystyle{\psi(x)=\sum_{j=1}^Nc_jf_j(x)}$.
• Using Gaussian quadrature to estimate the integrals got by multiplying the above by $f_i(x)$ and integrating, we get $\displaystyle{\frac{-\hbar^2}{2m}\sum_j (T_{ij}+V_{ij})c_j = E_i c_i }$, where $\displaystyle{V_{ij}\approx V(x_i)\delta_{ij}}$ is the potential energy matrix element and $\displaystyle{T_{ij}\approx -\lambda_i^{1/2} f_j''(x_i)}$, 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: $F_{ijk}(x,y,z)=f_i(x)g_j(y)h_k(z)$.

References:

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