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The Hypernetted-chain approximation used in statistical mechanics. Was for instance used in the theory of the fractional quantum hall effect by Laughlin in order to estimate the energies of elementar …

Proof of the existence of $\Omega$ : It is easy to see that the number operators $N_i = A_i^* A_i$ commute. Therefore there exists an orthonomal basis of common eigenvectors. Now let $\Omega$ one of t …

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator). Is this also true in the Solovay model …

I recommend the following problem : Proof the existence of the spectral gap for the fractional quantum Hall effect at least for a simplified model where Laughlin's wavefunction is the exact ground st …

In the BCS-theory of superconductivity the energy gap that seperates the ground state from the excited states is a non-analytic function of the exchange energy. Here one doesn't use a taylor expansio …

For VBS quantum antiferromagnets in one dimension see also : Ian Affleck, Tom Kennedy, Elliott H. Lieb and Hal Tasaki, Valence bond ground states in isotropic quantum antiferromagnets. Comm. Math. Ph …

Let $q(x,y) = x^H J y$ for $x,y \in \mathbb{C}^{2n}$ where $J = diag(I_n,-I_n)$ and let $S = \{A \in M_{2n}(\mathbb{R}) : q(Ax, Ax) \ge q(x,x) $ $\forall x \in \mathbb{C}^{2n}\}$. Obviously $S$ is a s …

For a list of 15 open problems in mathematical physics (in 2000) see Simon's Problems, http://mathworld.wolfram.com/SimonsProblems.html. Some of these problems are solved, as mentioned in the link.