Antisymmetric functions as Slater determinants Can any antisymmetric function, i.e., a function of $N$ spatial-plus-spin variables
$\{x_i\ | \ i= 1, \ldots, N\}$ satisfying
$$ \psi(x_1,\ldots, x_i, \ldots, x_j, \ldots, x_N) =  -\psi(x_1,\ldots, x_j, \ldots, x_i, \ldots, x_N)\ ,$$
always be written as a Slater-Determinant of functions ("orbitals"), not necessarily solutions to Schrödinger's Equation? (I know that there are no orbitals describing interacting electrons, but they may [or may not] be solutions to a "Schrödinger-like equation, but that's beside the point.)  
(As context: I have been reading up on Density Functional Theory. I am teaching an optional second course on quantum mechanics and I am explaining some of DFT's basic concepts. In Parr & Yang's "Density-functional theory of atoms and molecules" (Oxford University Press, 1989) it is mentioned that there is a proof that any "reasonable" (electron) density can be derived from an antisymmetric wave function which can always be written as Slater-Determinant of orbitals. The authors cite a couple of articles on some journals to which we, at the university where I work, have no subscription [Phys. Rev. A & B, Int. J. Quantum Chem.] )
 A: The short answer is: No, it is not true without other strong hypotheses.
What it is true is that any completely antisymmetric wavefunction $\psi(x_1,\ldots,x_N) \in L^2(\mathbb R^{3N})$ (not necessarily solution of Schroedinger equation) can always be written as a, generally infinite, linear combination of Slater determinants. 
Indeed, if $\{\phi_k\}_{k=1,2\ldots,}$ is a Hilbert basis of $L^2(\mathbb R^3)$ and $\phi \in L^2(\mathbb R^{3N})$ then:
$$\phi(x_1,\ldots,x_N) = \sum_{i_1,\ldots, i_N} C_{i_1...i_N} \phi_{i_1}(x_1)\cdots \phi_{i_n}(x_N)$$ 
where the convergence is that in $L^2$. Then consider the orthogonal projector  $A$ from $L^2(\mathbb R^{3N})$ onto the subspace of completely antisymmetric wavefunctions. if $\phi$ is generic, $\psi = A\phi$ is the generic completely antisymmetric wavefunction, so  we have that any completely antisymmetric wavefunction of $N$ entries can be decomposed as:
$$\psi(x_1,\ldots, x_N) = \sum_{i_1,\ldots, i_N} C_{i_1...i_N} A(\phi_{i_1}(x_1)\cdots \phi_{i_n}(x_N)) \:.$$
Above
$$A(\phi_{i_1}(x_1)\cdots \phi_{i_n}(x_N))$$ 
is nothing but the Slater determinant of $\phi_{i_1}(x_1)\:,\ldots\:, \phi_{i_n}(x_N)$.
The generalization to the case where $x_k$ includes spin variables is obvious.
