Symmetric and antisymmetric wave function

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Hallo !
Unfortunately, in general, neither symmetric nor anti-symmetric wavefunctions can be said to be eigenfunctions of the Hamiltonian.
The wave function for an electron in a hydrogen-like atom with atomic number Z in the ground state
is
RZ(r) is an eigenfunction of HZ=1/(2m)*p^2-Ze^2/(4πε0r).
But RZ(r) is not an eigenfunction of HZ'=1/(2m)*p^2-Z'e^2/(4πε0r), Z'≠Z.
Let us consider the case where a hydrogen-type atom with atomic number Z and a hydrogen-type atom with atomic number Z' are sufficiently separated from each other. And each electron in each atom is in the ground state.
The anti-symmetric wave function
is not an eigenfunction of the Hamiltonian
It should be an ironclad rule of quantum mechanics that the wave function is an eigenfunction of the Hamiltonian.

岡安一壽-gy