Consider solutions of the time-independent one-dimensional Schrödinger equation, in atomic units:

.

One trick for finding quasi-exact solutions is to assume some appropriately behaved function and to use the relation

to identify a potential function . For example, the simplest case of a quasi-exact sextic anharmonic oscillator follows from

,

which gives

,

with

.

More generally, it can be shown that the potential can have the form

, .

For the cases , and , we show plots of the quasi-exact eigenfunctions and of the potential functions , with superposed eigenvalues shown in red.