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Supersymmetric quantum mechanics is one of the most important branches, providing an important method for the rapid and convenient solution of the Schrödinger equation. For the trigonometric Scarf potential, we find that the shape invariance with two parameters shows a new characteristic, i.e., two parameters’ cross-additivity ( A1→B0+α/2, B1→A0+α/2). That is different with the parameters’ change ( A1→A0+α/2, B1→B0+α/2). The changing of the parameters with cross-additivity brings new characteristic to the wave function and energy spectrum. Based on this cross-additivity characteristic, we discuss in detail the eigenvalues and the eigenfunctions of the Hamiltonian with this potential. And then we get the two-parameter cross-additivity shape invariance again with potential algebra methods and study the energy spectrum. It is shown that the two-parameter cross-additivity shape invariance of the partner potential is completely self-consistent with its potential algebraic form. Our research indicates that the Schrödinger equation with a superpotential with two parameters shows new characteristics.

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In atomic and molecular physics, the Pöschl-Teller potential and its modified form (hyperbolic Pöschl-Teller potential) are particularly significant potentials. It is of great importance to study the Schrödinger equation with those potentials. In this paper, we further extend the hyperbolic Pöschl-Teller potential through generalizing the superpotential of that potential of the form Atanh (αx)-Bcoth (αx) to the more general form -Atanh (npx)-Bcoth (mpx). First, we introduce briefly the shape invariance and the potential algebra in supersymmetric quantum mechanics. Second, we derive three additive shape invariances, which are related to parameters A   and B of the partner potentials with the generalized superpotential, and discuss the eigenfunctions and eigenvalues in detail. Although the superpotential has two parameters, those shape invariances still belong to the one-parameter form. The reason is that there is always a constraint relationship between A and B in the additive shape invariance of the partner potentials. Third, through the potential algebra approach, we obtain the relevant shape invariance and calculate the corresponding eigenvalue of the Schrödinger equation with the potential of the generalized superpotential. The calculation shows that the algebraic form shape invariance of the partner potentials with that superpotential is anastomotic to the above. Last, we make a summary and outlook.