We mainly explore the linear algebraic structure like SU(2) or SU(1,1) of the shift operators for some one-dimensional exactly solvable potentials in this paper. During such process, a set of method based on original diagonalizing technique is presented to construct those suitable operator elements, J0, J_pm that satisfy SU(2) or SU(1,1) algebra. At last, the similarity between radial problem and one-dimensional potentials encourages us to deal with the radial problem in the same way.
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