The Lagrange-mesh method approximates a differential operator eigenvalue problem by combining an interpolation vector basis with Gaussian quadrature. Each interpolation function vanishes at all nodes except one. Using the same quadrature for the matrix elements produces a simple discrete eigenvalue problem (Baye and Heenen 1986, Baye et al. 2002).
For a one-dimensional Schrödinger operator, consider
 |
(1)
|
Take 
trial functions 
satisfying the boundary
conditions and the nodal conditions
 |
(2)
|
where 
are the positive quadrature weights for integration with
respect to 
, and 
is the Kronecker delta.
A weighting function can be absorbed into the 
and the 
. The quadrature approximation
to the Gram matrix is the identity
matrix. With 
, the
derivative term has approximate matrix elements
 |
(3)
|
Multiplication by 
is approximated by the diagonal matrix with entries
, giving
 |
(4)
|
The resulting eigenvalue problem is
 |
(5)
|
This is closely related to discrete variable representation.
For Hermite-Gauss quadrature, let 
be the zeros
of the Hermite polynomial 
and let 
be the quadrature weights
for 
. An exactly orthonormal vector basis is
 |
(6)
|
where 
, and the value at 
is defined by its limit. Its
derivative matrix has elements
 |
(7)
|
and, for 
,
 |
(8)
|
Thus the matrix is symmetric. Trott (2006) uses this type of interpolation vector basis to approximate the quartic oscillator.
The figure above shows the three interpolation functions for 
. Vertical guides mark the nodes 
, 0, and 
. Each function
vanishes at the other two nodes. The colored dots mark the nonzero nodal values 
.
Replacing exact integrals by quadrature generally removes the upper-bound property of the Rayleigh-Ritz
variational technique. For example, the two-node Hermite mesh for 
gives
 |
(9)
|
Its lowest eigenvalue is 
, below the quartic
oscillator's ground value 
. In contrast,
the Rayleigh-Ritz variational technique
on the same two-dimensional trial subspace with exact
integrals gives the upper bound 
.
