A quartic oscillator is the eigenvalue problem for a one-dimensional Schrödinger operator with a confining quartic polynomial potential. For the pure quartic oscillator, one normalization is
 |
(1)
|
where 
and the eigenfunction 
is square-integrable
on the real line. The self-adjoint
realization of this differential operator
has simple eigenvalues 
tending to infinity. The lowest eigenfunction
is the ground state.
The figure above shows the potential 
, the first four eigenvalues,
and their normalized eigenfunctions.
Replacing 
by 
with 
and putting 
gives the scaling law
 |
(2)
|
A quadratic term can also be included, giving the potential 
with 
.
The pure quartic oscillator is a useful test problem for numerical spectral methods (Hioe and Montroll 1975, Macfarlane 1999, Trott 2006). Its lowest eigenvalue is
 |
(3)
|
A simple upper bound follows from the normalized Gaussian function used as a trial function 
with 
. Its Rayleigh quotient
is
![R[psi_a]=a/2+3/(4a^2).](/images/equations/QuarticOscillator/NumberedEquation4.svg) |
(4)
|
Minimizing at 
gives
 |
(5)
|
For the Rayleigh-Ritz variational technique, use the normalized functions
 |
(6)
|
where 
is a Hermite polynomial and 
is a nonnegative integer.
The matrix has elements 
. These vanish unless
is 0, 2, or 4. The nonzero elements are
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
with 
(Trott 2006). Truncating this symmetric matrix
to a finite trial subspace gives upper bounds on the
eigenvalues. The even and odd index blocks can be
diagonalized separately because the potential is an even
function.
An alternative discretization uses the Lagrange-mesh method with interpolation functions based on the zeros of a Hermite polynomial. Hermite-Gauss quadrature approximates the matrix elements, and the potential is represented by a diagonal matrix of its values at the nodes (Baye et al. 2002, Trott 2006).
A complex extension uses 
with 
(Trott 2006). It satisfies 
, the potential condition for parity-time
symmetry. For 
,
this differential operator is not self-adjoint
in the usual inner product, so the preceding self-adjoint
Rayleigh-Ritz variational technique
does not automatically apply.
