Küttler, Heinrich (2014): Anderson's orthogonality catastrophe. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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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. 18:10491051], 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 finiterange 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 finiterange 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. 329:979998], 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. 18:10491051]. Another publication by Anderson from the same year, [Phys. Rev. 164:352359], 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. 164:352359], and improves upon the one in [Commun. Math. Phys. 329:979998]. We use the methods from [Commun. Math. Phys. 329:979998], 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 coauthored with Martin Gebert, Peter Müller, and Peter Otte.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  Schrödinger operators, spectral theory, Fermi gas 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  19. September 2014 
1. Referee:  Müller, Peter 
MD5 Checksum of the PDFfile:  ecbfb0422f3bdb2ac08a1ef8f054e8a8 
Signature of the printed copy:  0001/UMC 22373 
ID Code:  17442 
Deposited On:  06. Oct 2014 08:35 
Last Modified:  23. Oct 2020 23:03 