Gebert, Martin (2015): Spectral and Eigenfunction correlations of finitevolume Schrödinger operators. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two noninteracting fermionic systems which differ by an exterior shortrange scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:10491051] and is used to explain anomalies in the Xray absorption spectrum of metals. We call this scalar product $S_N^L$, where $N$ refers to the particle number and $L$ to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension $d\in\N$, i.e. $S_N^L^2\le L^{\gamma}$ in the thermodynamic limit $N/L^d\to \rho>0$ approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsinT/2}_{\text{HS}}$, where T refers to the scattering Tmatrix. In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the $3$dimensional toy model of a Dirac$\delta$ perturbation that the exact decay exponent is given by $\zeta:= \delta^2/ \pi^2$. Here, $\delta$ refers to the swave scattering phase shift. In particular, this result shows that the previously found decay exponent $\gamma$ does not provide the correct asymptotics of $\S_N^L$ in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the nonvanishing of the expectation value of the noninteracting manybody scalar product in the thermodynamic limit. Apart from the behavior of the scalar product of the noninteracting ground states, we also study the asymptotics of the difference of the groundstate energies. We show that this difference converges in the thermodynamic limit to the integral of the spectralshift function up to the Fermi energy. Furthermore, we quantify the error for models on the halfaxis and show that higher order error terms depend on the particular thermodynamic limit chosen.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  Schrödinger operators, spectral theory, Fermi projection 
Subjects:  600 Natural sciences and mathematics 600 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  23. September 2015 
1. Referee:  Müller, Peter 
Persistent Identifier (URN):  urn:nbn:de:bvb:19187248 
MD5 Checksum of the PDFfile:  e3ed47d7e35aca3ff8e1e1eab25179ab 
Signature of the printed copy:  0001/UMC 23289 
ID Code:  18724 
Deposited On:  21. Oct 2015 13:32 
Last Modified:  13. Jan 2016 10:05 