Maslov's quantization conditions for the bound states of the hydrogen atom (Q1819800)

It is shown that Maslov's quantization condition determines the eigenvalues of the Schrödinger operator \(\hat H\) of the hydrogen atom, the angular momentum \({\hat \ell}_ 1\) and the Lenz operator \(\hat e{}_ 1\). One investigates eigenfunctions of \(\hat H\) which are also eigenfunctions of \({\hat \ell}_ 1\) and \(\hat e{}_ 1\), and the multiplicities of the eigenspaces for the hydrogen atom are determined. The Hamiltonian functions of \(\hat H\), \({\hat \ell}_ 1\), and \(\hat e{}_ 1\) are introduced and the associated level set L(E,\({\bar \ell}_ 1,\bar e_ 1)\) is defined. Then, it is proven that L(E,\({\bar \ell}_ 1,\bar e_ 1)\) satisfies Maslov's quantization conditions if and only if E, \({\bar \ell}_ 1\), and \(\bar e{}_ 1\) are eigenvalues of the operators \(\hat H\), \({\hat \ell}_ 1\) and respectively \(\hat e{}_ 1\). It is also shown that for each eigenvalue \(E=E_ n\) of \(\hat H\) the number of \(L(E_ n,{\bar \ell}_ 1,\bar e_ 1)\) satisfying the quantization condition is equal to the multiplicities of the eigenspaces of \(\hat H\) corresponding to \(E_ n\). It is concluded that an eigenvalue problem can be solved without using operator theory, but considering classical mechanics, the invariant Lagrangian submanifolds and Maslov's quantization condition.