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I think this question belongs to mathematical physics. The Hamiltonian of an N-electron atom in a homogeneous electric field is

$$ H =\left( \sum_{i=1}^N \frac{p_i^2}{2m } - \frac{Z e^2}{r_i} - E_z e z_i \right) + \sum_{i\neq j } \frac{e^2}{r_{ij}} . $$

Let us denote the ground state energy as a function of the external electric field as $E(E_z)$.

Now, the conjecture is that, $E_z =0 $ is a minimum. It can never be a maximum.

This is quite obvious by the physical intuition. But, how to prove it?

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    $\begingroup$ The operator is unbounded below, there is no "ground state energy." $\endgroup$
    – Christian Remling
    May 22, 2016 at 18:27

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