  We make use of a recently developed method to, not only obtain the exactly
known eigenstates and eigenvalues of a number of quasi-exactly solvable
Hamiltonians, but also construct a convergent approximation scheme for locating
those levels, not amenable to analytical treatments. The fact that, the above
method yields an expansion of the wave functions in terms of corresponding
energies, enables one to treat energy as a variational parameter, which can be
effectively used for the identification of the eigenstates. It is particularly
useful for the quasi-exactly solvable systems, where the ground state is known
and a number of eigenstates are bounded, both below and above. The efficacy of
the procedure is illustrated by obtaining, the low-lying excited states of a
prototypical double-well potential, where the conventional techniques are not
very reliable. Our approach yields the approximate eigenfunctions and
eigenvalues, whose accuracy can be improved to any desired level, in a
controlled manner. Comparing the present results with those of an independent
numerical method, it was found that, the first few terms in our approximate
solutions are enough to yield the excited state eigenvalues, accurate upto the
third place of the decimal.
