On an equivalent Hermitian Hamiltonian for the wrong-sign quartic potential

In this paper I present the results of a calculation with J. Mateo, in which, by a judicious choice of the contour on which the Schrödinger equation for the potential ?gz ⁴ is posed, we were able to give an explicit construction of an equivalent Hermitian Hamiltonian with the same spectrum. I also discuss the functional-integral approach to constructing equivalent Hamiltonians. In many cases this gives the simplest derivation. However, in this particular case it only gives the classical Hamiltonian, without a linear term, which is in fact an anomaly that can only be obtained by a careful discretization of the functional integral.