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 
.
