Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential

Abstract

In this paper, we search the factorizations of the shape invariant Hamiltonians with Scarf II potential. We find two classes: one of them is the standard real factorization which leads us to a real hierarchy of potentials and their energy levels; the other one is complex and it leads us naturally to a hierarchy of complex Hamiltonians. We will show some properties of these complex Hamiltonians: they are not parity-time (or PT) symmetric, but their spectrum is real and isospectral to the Scarf II real Hamiltonian hierarchy. The algebras for real and complex shift operators (also called potential algebras) are computed; they consist of su(1, 1) for each of them and the total potential algebra including both hierarchies is the direct sum su(1, 1) ⊕ su(1, 1).

Unified Supersymmetric Description of Shape-Invariant Potentials within and beyond the Natanzon Class

“…Further findings will be discussed in detail elsewhere, leaving the present work focusing on real potentials only. For the general complex Scarf II potential, see, e.g., the recent study [37].…”

Optical ladder operators in the Glauber-Fock oscillator array

“…Besides, due to the formal equivalence between the Schrödinger equation and the paraxial Helmholtz equation [21] (also between the stationary Schrödinger equation and the Helmholtz equation), optical waveguides are suitable devices to observe, study and test quantum phenomena [22]. Therefore, either isolated waveguides or waveguide arrays (optical lattices), are susceptible to be transformed by means of Darboux or SUSY transformations [23][24][25][26][27][28][29][30][31][32][33][34], in the Hermitian [26][27][28][31][32][33][34] and non-Hermitian [9,10,[23][24][25][28][29][30] regimes. Specifically, an optical lattice associated with a Hamiltonian of the type of the quantum harmonic oscillator can be 'intertwined' with itself, giving rise to the ladder of supermodes (eigenstates) [35,36].…”