  The potential -x^4, which is unbounded below on the real line, can give rise
to a well-posed bound state problem when x is taken on a contour in the
lower-half complex plane. It is then PT-symmetric rather than Hermitian.
Nonetheless it has been shown numerically to have a real spectrum, and a proof
of reality, involving the correspondence between ordinary differential
equations and integral systems, was subsequently constructed for the general
class of potentials -(ix)^N. For PT-symmetric but non-Hermitian Hamiltonians
the natural PT metric is not positive definite, but a dynamically-defined
positive-definite metric can be defined, depending on an operator Q. Further,
with the help of this operator an equivalent Hermitian Hamiltonian h can be
constructed. This programme has been carried out exactly for a few soluble
models, and the first few terms of a perturbative expansion have been found for
the potential m^2x^2+igx^3. However, until now, the -x^4 potential has proved
intractable. In the present paper we give explicit, closed-form expressions for
Q and h, which are made possible by a particular parametrization of the contour
in the complex plane on which the problem is defined. This constitutes an
explicit proof of the reality of the spectrum. The resulting equivalent
Hamiltonian has a potential with a positive quartic term together with a linear
term.
