Class A has the advantage that the -independent basis preserves
the linearity of the eigenproblem, giving a matrix eigenequation which
returns many solutions
at once.
This is the conventional `basis diagonalization' approach.
Generally *all* the eigenstates from 1 up to a maximum useable state
are returned. Clearly cannot be less than this number of states.
For instance in the case of a dimensional billiard, this means must
scale like if states at wavenumber are sought. This is clearly
a huge limitation.

The basis can be chosen as the analytically-known eigenstates of a simpler Hamiltonian ; this I call Class A1. The resulting basis is therefore orthonormal and complete (in the limit) and if is `close' to then will not need to be much higher than , where is the typical quantum number of the desired states at energy .

Alternatively the basis is chosen to be convenient in position space (or
momentum space, or a mixture of both). The basis is
effectively complete (up to energies of interest) because it entirely covers
the domain .
This I call Class A2.
The advantage of these localized basis functions is that the resulting
matrix is *sparse*, allowing much faster extraction of eigensolutions.
Various smoothnesses of such basis functions are possible, ranging from lattices
(corresponding to piece-wise linear `pyramidal' functions)
through other Finite Element [87,19,182,56] basis functions
and higher-order spline functions [70], to gaussian packets
(coherent states, or the Distributed Gaussian
Basis [DGB][57]).
The smoother a basis is, the faster the convergence with can be
for a given energy of interest.
However, smooth basis functions are more complicated to
construct (especially if definite BCs are required), to evaluate,
and result in less sparse matrices.
Lattice methods (often known as `finite differencing')[161,18]
generate sparse matrices which can be diagonalized much faster than
dense ones, but errors converge only like a power law
.
One smooth basis with useful sparsity properties is the Discrete-Variable
Representation (DVR) [17,86,180].
Most methods involve a compromise.
The optimal basis set choice for smooth-potential problems
appears to be a covering of phase space by gaussian packets, in which
case need only be a couple of times larger than
(for small dimensions ) [57,138].
Because the Wigner function for such problems dies exponentially
outside the classically-allowed region of phase space, these
phase-space covering methods achieve exponential convergence with ,
once is larger than the semiclassical basis size.

Note that if the basis is not orthogonal, as is frequently the case for Class A2 basis sets, the eigenequation becomes a generalized eigenequation where the matrix gives the basis function overlaps.