Lagrange Mesh Method in Quantum Mechanics

• Lagrange Mesh Method is a computational technique leveraging mesh collocation and Gauss quadrature to solve quantum eigenvalue and reaction equations with high accuracy.
• It uses specialized Lagrange functions that vanish at all but one mesh point, simplifying matrix assembly with diagonal potential and overlap matrices.
• The method is versatile in addressing local and nonlocal operators across nuclear, atomic, and molecular physics, including extensions to relativistic and scattering problems.

The Lagrange Mesh Method is a computational technique for solving quantum mechanical eigenvalue problems and reaction equations. It is designed around the use of mesh-based collocation points and associated quadrature rules, providing both the simplicity of grid-based calculations and the accuracy of variational methods. The approach is built on specialized basis sets—Lagrange functions—defined to vanish at all but one mesh point and to be orthonormal under a quadrature rule exact for polynomials up to a degree determined by the mesh size and weight. Applications span nuclear and atomic physics, including configuration space, momentum space, relativistic treatments, and scattering/reaction formalisms.

1. Foundations: Lagrange Functions and Gauss Quadrature

The cornerstone of the method is the construction of Lagrange functions $f_i(x)$ tailored to a mesh and quadrature pair. For a set of mesh points—typically the zeros of an orthogonal polynomial (Legendre, Laguerre, Hermite, Jacobi, etc.)—and associated weights $\lambda_i$, the interpolation property is

$f_j(x_i) = \lambda_i^{-1/2}\, \delta_{ij}$

with

$\int_a^b f_i(x) f_j(x) dx \approx \delta_{ij}$

under the Gauss quadrature rule. These functions act as a discrete-variable representation (DVR) basis. For example, on $[0,\infty)$, one uses regularized Laguerre-based functions, while on finite intervals $[0,a]$, the Legendre-based forms are common (Baye et al., 2014, Lacroix et al., 2011, Shubhchintak et al., 2022, Dohet-Eraly, 2016).

The basis expansion for a wavefunction is

$\psi(x) = \sum_{j=1}^N c_j\,f_j(x)$

yielding a matrix problem whose potential is diagonal and whose kinetic terms can be calculated analytically in terms of mesh points and weights.

2. Matrix Elements: Local and Nonlocal Operators

For any local operator $V(x)$, the matrix elements in the Lagrange basis are diagonal at the Gauss approximation: $$\langle f_i|V|f_j \rangle \approx V(x_i)\, \delta_{ij}$$ The calculation is thus reduced to evaluating the potential at the mesh points (Baye et al., 2014, Lacroix et al., 2011, Shubhchintak et al., 2022, Dohet-Eraly, 2016).

For nonlocal operators (e.g., Perey–Buck type nuclear potentials), the matrix elements are

$$\iint f_i(r) v(r,r') f_j(r') dr dr' \approx \sqrt{\lambda_i\lambda_j} v(r_i, r_j)$$

This enables the efficient treatment of nonlocal effects in nuclear or atomic interactions (Phuc et al., 2021).

For kinetic energy (second derivatives), closed-form expressions for matrix elements are available: $$T_{ij} = \int f_i(x) \left(-\frac{d^2}{dx^2}\right) f_j(x) dx$$ For the Laguerre mesh,

$$T_{ij} = \begin{cases} (-1)^{i-j}\sqrt{x_i x_j}/(x_i - x_j)^2 & i \ne j \ \frac{1}{12} - \frac{1}{2 x_i} & i = j \end{cases}$$

More general backgrounds (e.g., relativistic Dirac equations) use similar principles with modified basis constructions, as in Dirac–Lagrange mesh implementations (Baye et al., 2014, Filippin et al., 2015, Filippin et al., 2016).

3. Lagrange-Mesh R-Matrix Formalism

For reaction and scattering problems, the Lagrange mesh is utilized within the $R$-matrix formalism, which divides configuration space at a channel radius $a$:

• Internal region ($r \leq a$): wave functions are expanded in the Lagrange basis.
• External region ($r > a$): matching to known asymptotics (Coulomb, Whittaker, etc.) is performed.

The Bloch surface operator is incorporated to restore Hermiticity: $$\mathscr{L}(B) = \frac{\hbar^2}{2\mu} \delta(r - a) \left( \frac{d}{dr} - \frac{B}{r} \right)$$ The combined Hamiltonian and Bloch operator yields a matrix $C(E,B)$. After inversion, the $R$-matrix is

$$\mathscr{R}_{\ell j}(E) = \frac{\hbar^2}{2\mu a} \sum_{i,n=1}^N \varphi_i(a) [C^{-1}(E,0)]_{in} \varphi_n(a)$$

Phase shifts and $S$-matrix elements are extracted by matching derivatives at the channel radius (Phuc et al., 2021, Shubhchintak et al., 2022, Lei et al., 2020, Shubhchintak et al., 2019).

For inhomogeneous equations, as arise in breakup and transfer reactions with source terms: $[T + U - E] u(r) = \rho(r)$ the solution proceeds via expansion over the mesh and inversion of the modified system (Lei et al., 2020).

4. Momentum-Space and Fourier Transform Extensions

The Lagrange Mesh Method applies in momentum space using a Gauss–Laguerre mesh for $p \in [0,\infty)$, with analogous regularized functions $f_i(p/h)$, where $h$ is a momentum-scale parameter. The kinetic operator $T(p^2)$ is diagonal in this basis: $T_{ij} = T(p_i^2)\, \delta_{ij}$ For nonlocal and singular potentials, two strategies are employed:

• Direct quadrature using partial-wave Fourier transforms of $V(r)$.
• $r^2$-diagonalisation: construct $r^2_{ij}$ matrix, diagonalize, evaluate $V$ on its spectrum, and transform back; this generalizes to Coulomb, linear, and other long-range potentials without encountering divergences (Chevalier et al., 3 Oct 2025, Lacroix et al., 2012).

Observables in momentum space and configuration space are accessible through this framework, typically via expressions such as

$\langle U(p) \rangle = \sum_{i} C_i^2 U(p_i)$

Wavefunctions in $r$ and $p$ spaces are related by mesh-based Fourier transforms (e.g., via spherical Bessel $j_\ell$ integrals at mesh points) (Lacroix et al., 2011, Lacroix et al., 2012, Chevalier et al., 3 Oct 2025).

5. Relativistic Effects and Regularization

For Dirac and relativistic Schrödinger equations (central atomic and nuclear potentials), the Lagrange–Laguerre mesh is adapted by selecting parameter $\alpha$ in $L_N^{(\alpha)}(x)$ to match the small-$r$ asymptotic behavior. This enables numerically exact energies (even for highly singular $1/r^2$ or $1/r$ potentials) with minimal mesh sizes, especially when regularized forms such as $\sqrt{r}$-regularized basis functions are employed (Baye et al., 2014, Filippin et al., 2015, Filippin et al., 2016, Dohet-Eraly, 2016).

For physical observables such as polarizabilities and two-photon transition rates, three meshes (for initial, final, and intermediate states) greatly enhance accuracy in scenarios with differing small-$r$ behavior (Filippin et al., 2016, Filippin et al., 2015).

6. Advanced Applications: Reaction Theory, Nuclear Matter, and Many-Body Systems

The method is integrated into:

• Nuclear transfer and direct-capture reaction theory, providing efficient and accurate DWBA and ADWA calculations, particularly when nonlocality and breakup are important (Phuc et al., 2021, Shubhchintak et al., 2022, Shubhchintak et al., 2019).
• Continuum-discretized coupled-channels (CDCC) calculations for breakup reactions, where the Lagrange mesh simplifies the discretization of the projectile continuum and accelerates coupled-channel equation integration (using enhanced Numerov algorithms) (Chen et al., 2021).
• Self-consistent mean-field approaches for nuclear slab geometries, using the Lagrange mesh to handle finite-range nonlocal Fock terms and large boxes, minimizing Friedel oscillations and improving surface energy accuracy compared to Numerov-type algorithms (Davesne et al., 12 Nov 2025).
• Multi-body quantum dot calculations, confined atomic and molecular systems, and general few-body Coulomb problems using product meshes in multiple dimensions (Baye et al., 2015, Pilón et al., 2014, Hiltunen et al., 2013, Pilón, 2011).

The mesh approach often achieves high accuracy—up to 8–15 significant digits—in energies, wavefunctions, and observables due to the exactness of the quadrature and the optimal basis regularization.

7. Numerical Properties, Convergence, and Practical Implementation

Key numerical properties include:

• Exponential convergence with mesh size $N$ when the mesh and scale parameters are well chosen to cover the extent of the physical wavefunctions (Baye et al., 2014, Phuc et al., 2021, Lacroix et al., 2011).
• Diagonal potential and overlap matrices under quadrature lead to sparse, efficiently diagonalized Hamiltonians.
• For nonlocal potentials or coupling kernels, only a single $N \times N$ matrix inversion is generally required (Phuc et al., 2021, Lei et al., 2020).
• Sensitivities associated with singular potentials or large boxes are controlled via basis regularization and the choice of mesh scale (Dohet-Eraly, 2016, Davesne et al., 12 Nov 2025).

Implementation typically consists of:

• Calculation of mesh points and weights corresponding to the chosen quadrature/polynomial.
• Construction of regularized Lagrange functions.
• Assembly of Hamiltonian, potential, and kinetic matrices by quadrature and analytical formulas.
• Diagonalization to obtain energies and expansion coefficients.
• Evaluation of observables and, if needed, Fourier transforms or reaction matrix elements via mesh sums.

The Lagrange Mesh Method’s performance is consistently superior to conventional finite-difference or finite-element approaches for quantum mechanical problems where analytic basis regularity and quadrature-collapsing lead to optimal accuracy and efficiency. Adoption continues to expand in precision atomic, molecular, and nuclear structure and reaction theory.