Abstract

The topic of this thesis is a mathematical treatment of Anderson's orthogonality catastrophe. Named after P.W. Anderson, who studied the phenomenon in the late 1960s, the catastrophe is an intrinsic effect in Fermi gases. In his first work on the topic in [Phys. Rev. Lett. 18:1049--1051], Anderson studied a system of $N$ noninteracting fermions in three space dimensions and found the ground state to be asymptotically orthogonal to the ground state of the same system perturbed by a finite-range scattering potential. More precisely, let $\Phi_L^N$ be the $N$-body ground state of the fermionic system in a $d$-dimensional box of length $L$,and let $\Psi_L^N$ be the ground state of the corresponding system in the presence of the additional finite-range potential. Then the catastrophe brings about the asymptotic vanishing $$\S_L^N := \<\Phi_L^N, \Psi_L^N\> \sim L^{-\gamma/2}$$ of the overlap $\S_L^N$ of the $N$-body ground states $\Phi_L^N$ and $\Psi_L^N$. The asymptotics is in the thermodynamic limit $L\to\infty$ and $N\to\infty$ with fixed density $N/L^d\to\varrho > 0$. In [Commun. Math. Phys. 329:979--998], the overlap $\S_L^N$ has been bounded from above with an asymptotic bound of the form $$\abs{\S_L^N}^2 \lesssim L^{-\tilde{\gamma}}$$. The decay exponent $\tilde{\gamma}$ there corresponds to the one of Anderson in [Phys. Rev. Lett. 18:1049--1051]. Another publication by Anderson from the same year, [Phys. Rev. 164:352--359], contains the exact asymptotics with a bigger coefficient $\gamma$. This thesis features a step towards the exact asymptotics. We prove a bound with a coefficient $\gamma$ that corresponds in a certain sense to the one in [Phys. Rev. 164:352--359], and improves upon the one in [Commun. Math. Phys. 329:979--998]. We use the methods from [Commun. Math. Phys. 329:979--998], but treat every term in a series expansion of $\ln S_L^N$, instead of only the first one. Treating the higher order terms introduces additional arguments since the trace expressions occurring are no longer necessarily nonnegative, which complicates some of the estimates. The main contents of this thesis will also be published in a forthcoming article co-authored with Martin Gebert, Peter Müller, and Peter Otte.