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 $su(1,1)\oplus su(1,1)$.

Figures (5)

• Figure 1: Plot of the real Scarf II potential for different values of parameters: $\alpha=3, \beta_1=3.3, \beta_2=5.3, \beta_3=7.3$ (left) and for $\beta=5.3, \alpha_1=1, \alpha_2=3, \alpha_3=5$ (right).
• Figure 2: Plot of the energies, the eigenfunctions of the real Scarf II potential for $\alpha=3, \beta=6.8$ and $n=0,1,2$ where $n_{max}=[(6.8-1)/2]=[2.9]=2$.
• Figure 3: Plot of the real part (left) and the imaginary part (right) of the complex Scarf II potential for different values of parameters: $\beta=5.3, \alpha_1=3, \alpha_2=3+2i, \alpha_3=3+4i$. These are the complex potentials of the complex hierarchy corresponding $\beta=5.3$.
• Figure 4: Plot of the solutions of the complex Scarf II potential for $\alpha=3+2i, \beta=6.8$ for $n=0,1,2$.
• Figure 5: Schematic drawing of hierarchies of the real and the complex Scarf II potential Hamiltonians.