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In the adiabatic approximation one looks at the Hamiltonian

$$ H_0 = \sum_{i = 1}^{N_e} \frac{\vec{p}_i^2}{2m_e} + \sum_{i < j} \frac{e^2}{|\vec{r}_i - \vec{r}_j|} + \sum_{k < l} \frac{Z_k Z_l e^2}{|\vec{R}_k - \vec{R}_l|} + \sum_{i, k} \frac{- Z_k e^2}{|\vec{r}_i - \vec{R}_k|}. $$

Now typical statements are

"For a fixed ion configuration $ \{\vec{R}_1,\ldots, \vec{R}_n\}$ let $\Psi_{\alpha} \equiv \Psi_{\alpha}(\vec{r}_1,\ldots,\vec{r}_n,\vec{R}_1,\ldots\vec{R}_n)$ be a solution of the eigenvalue problem $$ H_0 \Psi_{\alpha} = \varepsilon_{\alpha}(\vec{R}_1, \ldots \vec{R}_n) \Psi_{\alpha},$$ where $\alpha$ denotes a complete set of quantum numbers."

What is baffling to me is: How does one know about the existence of such solutions? (Not only, that one finds one solution of that eigenvalue problem, but also a complete set of solutions?!)

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Perhaps the answer may lie in properties of Hermitian operators. (The Hamiltonian is Hermitian). For example, the solutions (eigenfunctions) of a Hermitian operator form a complete orthogonal set. Another property of Hermitian matrices is that an $nxn$ Hermitian operator (matrix) has $n$ eigenvalues. So perhaps that answers why the solutions are guaranteed to exist.

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