Isospectral hermitian counterpart of complex nonhermitian Hamiltonian p2 – gx4 + a/x2

Abstract

We show that the nonhermitian Hamiltonians H = p2 – gx4 + a/x2 and the conventional hermitian Hamiltonians h = p2 + 4gx4 + bx [Formula: see text] are isospectral if a = (b2 – 4gℏ2)/16g and a ≥ –ℏ2/4. This new class includes the equivalent nonhermitian–hermitian Hamiltonian pair, p2 – gx4 and [Formula: see text] found by Jones and Mateo six years ago as a special case. When a = (b2 – 4gℏ2)/16g and a < –ℏ2/4 although h and H are still isospectral, b is pure imaginary, and h is no longer the hermitian counterpart of H.