A Special Case of a Conjecture by Widom with Implications to Fermionic Entanglement Entropy

We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semiclassical expansion of the trace of the square of certain integral operators on the Hilbert space $L^2(\mathbb R^d)$. As already observed by Gioev and Klich, this implies that the bipartite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.