The ladder operator method is used to solve the one-particle Schrodinger equation with a harmonic potential.
What other potentials for the one-particle Schrodinger equation may be solved with the ladder operator method?
Answer
The hydrogen atom is an example where ladder operators can be used. There is a hidden SO(4) symmetry that explains the degeneracy for the prinicpal quantum number and one can use algebraic methods to get the eigenvalues. Here is a paper that does central force problems in general. . The full symmetry group for the hydrogen atom is SO(4,2). Here is another resource.
Some people think ladder operators are only for the harmonic oscillator or equally spaced eigenvalues but this is because they are restricting themselves to the Heisenberg Lie algebra which works for the harmonic oscillator but there are other problems with other Lie algebras and their own representation theory.
For a trivial counter example for the belief you need equally spaced eigenvalues: Suppose your Hamiltonian was just $L^2$ . This is hermitian so legitimate, there is an algebraic method to solve for eigenvalues it depends on the representation theory of $su(2)$ or equivalently $sl(2,C)$ but as we all know the eigenvalues of this operator are not equally spaced.
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