Non-hermitian Hamiltonians and the Painleve IV equation with real parameters

In this Letter we will use higher-order supersymmetric quantum mechanics to obtain several families of complex solutions g(x;a,b) of the Painleve IV equation with real parameters a,b. We shall also study the algebraic structure, the eigenfunctions and the energy spectra of the corresponding non-hermitian Hamiltonians. -- Highlights: → We use higher-order supersymmetric quantum mechanics to derive deformed algebras. → We obtain complex solutions of the Painleve IV equation with real parameters. → We analyze the algebraic structure of the corresponding non-hermitian Hamiltonians. → We study the eigenfunctions and the energy spectra of non-hermitian Hamiltonians.