In what way is the potential $\lambda^2 / x^2 - \lambda/x$ different from the potential $1/x^2$

I am trying to determine the ground state energy and wave function for a particle of mass m confined to a one dimensional potential of the form $\lambda^2 / x^2 - \lambda/x$ . I found a paper that explains how the potential $1/x^2$ can be solved for the needed quantities, but i am confused how the first potential is different from the second. one thing is the first potential have ground state, while the second one has no ground state.

how is the solution to my question different from the solution to the 1/x^2 potential.

• highenergy.phys.ttu.edu/~akchurin/PHYS5302ProjectPapers/… when i read this paper, this is what i understood. If i am wrong please change them to the correct tags. Commented Jan 12, 2017 at 13:50
• It's not clear to me what your precise question is. Do you want someone to do an analysis similar to the paper you link of your modified potential for you? (That would be off-topic as homework-like in my eyes) Commented Jan 12, 2017 at 14:24