It can be proven that the commutation relations $$\Big[ N,H\Big]=0$$ $$\Big[ L,H\Big]=0$$ $$\Big[ N,L^2\Big] \neq 0$$ Where $N$ is the Runge-Lenz vector. I'm wondering if it is possible to write down the features of the spectrum of $H$ using the commutation relations?
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yes, Pauli did that in 1925. Google "hydrogen atom algebraic solution" – AccidentalFourierTransform Feb 05 '17 at 18:15
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Essentially a duplicate of http://physics.stackexchange.com/q/116244/2451 , http://physics.stackexchange.com/q/89654/2451 , http://physics.stackexchange.com/q/242035/2451 and links therein. – Qmechanic Feb 05 '17 at 18:24