Is there a "quantum state" of an individual electron in a multiple electron system

The Pauli exclusion principle claims that in a non-hydrogen atom, each electron occupies a separate "quantum state". Similarly, in a piece of metal, free electrons fill a "Fermi sea" of states, each falling to its own pigeonhole which is determined by the "quantum state" of the electron.

We criticized the Pauli exclusion principle because there is no definition of what that "quantum state" is, or means.

Let us look at the helium atom. The usual way of modeling it is to assume that we have a single particle moving in a 6-dimensional space. There is a central potential due to the nucleus and a "planar potential" which has a very high energy when the single particle position is equivalent to the two electrons being very close.

If we find a solution of the Schrödinger equation, is there any way to "factorize" it into two parts where each part describes the state of a single electron?

If we have two electrons in different hydrogen atoms, there exists such a factorization. It is trivial.

Let us look at a simpler problem. Suppose that we have two particles in an external potential well in a space with a time dimension and one space dimension. There is a repulsion between the particles. The Schrödinger equation then is about a single particle in a 1+2-dimensional space.

The external potential makes a square well for the single particle. In addition to that, there is the interaction potential of the original two particles. That potential is concentrated on the line y = x, where x and y are the spatial coordinates of the single particle.
                 ________
                |         / |
                |      /    |
                |   /       |
                |/______|
  ^ y
  |
  |
   ---------> x

The diagram is not in scale. The rectangle depicts the square potential well. The diagonal depicts the interaction potential wall of the original two particles.

A solution of the Schrödinger equation is a stationary wave inside the rectangle. It is like a rectangular drum skin vibrating in a resonance pattern within that square.

Now, is there any reason why the solution could be factorized into solutions of two individual particles?

If the particles do not interact, then the factorization is trivial. If there is a weak interaction, we may get some results with perturbation methods. But what if the interaction is strong?

Assume that each solution is determined uniquely by a set of quantum numbers of each electron

Let us assume that we have a strongly interacting electron system. Let us assume that each solution of the Schrödinger equation is uniquely (except by a phase factor) determined by some "quantum numbers" that we attach to each individual electron.

Can we derive the Pauli exclusion principle, for example, from the antisymmetricity of the fermion wave function? The antisymmetry means that the sign of the wave function is flipped if we replace coordinate values of, say, x_1, y_1, z_1 with x_2, y_2, z_2, and conversely.

Now, if electrons 1 and 2 have the exact same quantum numbers, we have:

       Ψ_switched = Ψ_original,

because the sequence of quantum numbers specifying Ψ did not change.

But, the antisymmetry of the fermion wave function implies

       Ψ_switched = -Ψ_original.

We have that Ψ must be zero. We can derive the Pauli exclusion principle from the assumptions:

1. The wave function solution is uniquely determined by a set of "quantum numbers" which can be "assigned" to the coordinate triplet of each electron.

2. The wave function is antisymmetric under the switch of two coordinate triplets.

Is there a mathematical proof that helium atom solutions have property 1 above?

A brief Internet search does not lead us to any such proof. A related question is in which cases a wave equation has a discrete spectrum of stationary states or "resonant" states.

https://en.wikipedia.org/wiki/Spectral_theorem

The spectral theorem states that all the solutions of the Schrödinger equation can be written as sums of eigenfunctions of the hamiltonian. Each eigenfunction is associated with an energy eigenvalue.

In which cases is the spectrum of energy eigenvalues discrete?

Does the spectral theorem imply anything about a multiple electron system? Could the theorem give some factorization of the solution for individual electrons?