Rayleigh-Ritz Variational Technique

The Rayleigh-Ritz variational technique approximates eigenfunctions and eigenvalues by restricting a variational problem to a finite-dimensional trial subspace. For a Sturm-Liouville equation, the variational formulation requires

(1)

to have a stationary value subject to the normalization condition

(2)

and the boundary conditions

(3)

This leads to the Sturm-Liouville equation

(4)

which gives the stationary values of

(5)

as

(6)

where are the eigenvalues corresponding to the eigenfunction .

For linearly independent trial functions satisfying the homogeneous boundary conditions , substitute

(7)

The Rayleigh quotient restricted to this subspace gives the generalized eigenvalue problem

(8)

where the matrices have elements

			(9)
			(10)

The Gram matrix is positive definite when . For an orthonormal basis in the weighted inner product, is the identity matrix. For a regular self-adjoint Sturm-Liouville equation with , the approximate eigenvalues, ordered increasingly and counted with multiplicity, are upper bounds on the corresponding exact eigenvalues. Enlarging a nested trial subspace can only decrease these upper bounds (Teschl 2009, pp. 118-119). These bounds require the exact trial-space matrix elements. Replacing them by quadrature approximations, as in the Lagrange-mesh method, need not preserve the upper-bound property.

The figure above uses the quartic oscillator and the nested trial subspaces spanned by the first even Hermite polynomials multiplied by Gaussian factors. The left panel shows upper bounds on the lowest eigenvalue. The right panel shows their distance from the high-precision reference value given by Trott (2006). The points are computed for , 2, 4, 8, 16, 32, and 64.