Karadzhov Y. Classification of shape-invariant Schroedinger equations

Invented by E. Witten as a toy model supersymmetric quantum mechanics (SSQM) became a fundamental field including many interesting external and internal problems. In particular the SSQM presents powerful tools for explicit solution of quantum mechanical problems using the shape invariance approach. Unfortunately, the number of problems satisfying the shape invariance condition is rather restricted. However, such problems include practically all cases when the related Schroedinger equation is exactly solvable and has an explicitly presentable potential. Well known exceptions are exactly solvable Schroedinger equations with Natanzon potentials which are formulated in terms of implicit functions. The list of shape invariant potentials depending on one variable can be found in the work of Cooper et al. Generalizing the supersymmetric PS problem we find a family of matrix potentials for Shroedinger equation satisfying the shape invariance condition. Let us stress that we present the completed classification of shape invariant superpotentials of the generic form $W_k=kQ+P+ 1/k R$ where $P, Q$ and $R$ are hermitian matrices of arbitrary finite dimension. It was found potential that is a generalized effective potential for the PS problem. Moreover, these potentials coincide for a particular value $mu=0$ of arbitrary parameter $mu$. However, if $mu neq0$ this potential is not equivalent to the potential appearing in the PS problem and corresponds to a more general interaction in the initial three-dimension problem. There where described an infinite number of shape-invariant integrable systems. In particular we present the list of superpotentials realized by matrices of dimension $2 times 2$. The main value of the list is its completeness, i.e., it includes all superpotentials realized by $2 times 2$ matrices which correspond to Schroedinger-Pauli systems being shape-invariant w.r.t. shifts of variable parameters. Superpotentials of this form can be used to describe problems of quantum mecanics for particles with spin $1/2$. For shape-invariant problems corresponding to $2 times 2$ matrices with matrix $Q$ proportional to the unit matrix, the spectral problem was solved. There were found eigen values and ground state, it was proved the square-integrability of excited states.