thesis published Sven Joscha Henheik author 31d731d7-d235-11ea-ad11-b50331c8d7fb0000-0003-1106-327X László Erdös supervisor GradSch department LaEr department Random matrices beyond Wigner-Dyson-Mehta project This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. In Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices. In order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. In Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. In Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. Additionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs). https://research-explorer.ista.ac.at/download/19540/19542/Henheik-Thesis_source_final.zip application/zip https://research-explorer.ista.ac.at/download/19540/19553/Henheik-Thesis-pdfa_FINAL.pdf application/pdfno https://research-explorer.ista.ac.at/download/19540/19615/Henheik-Thesis-Volume1_print.pdf application/pdf https://research-explorer.ista.ac.at/download/19540/19616/Henheik-Thesis-Volume2_print.pdf application/pdf Institute of Science and Technology Austria2025 eng 2663-337X 978-3-99078-057-210.15479/AT-ISTA-19540 720 https://research-explorer.ista.ac.at/record/14343 https://research-explorer.ista.ac.at/record/13317 https://research-explorer.ista.ac.at/record/11732 https://research-explorer.ista.ac.at/record/12184 https://research-explorer.ista.ac.at/record/14421 https://research-explorer.ista.ac.at/record/10623 https://research-explorer.ista.ac.at/record/18112 https://research-explorer.ista.ac.at/record/19001 https://research-explorer.ista.ac.at/record/10642 https://research-explorer.ista.ac.at/record/19545 https://research-explorer.ista.ac.at/record/19546 https://research-explorer.ista.ac.at/record/19550 https://research-explorer.ista.ac.at/record/19551 https://research-explorer.ista.ac.at/record/19552 https://research-explorer.ista.ac.at/record/14542 https://research-explorer.ista.ac.at/record/17049 https://research-explorer.ista.ac.at/record/18764 https://research-explorer.ista.ac.at/record/19547 https://research-explorer.ista.ac.at/record/19548 https://research-explorer.ista.ac.at/record/18656 S.J. Henheik, Modeling Complex Quantum Systems: Random Matrices, BCS Theory, and Quantum Lattice Systems, Institute of Science and Technology Austria, 2025. Henheik, Sven Joscha. <i>Modeling Complex Quantum Systems: Random Matrices, BCS Theory, and Quantum Lattice Systems</i>. Institute of Science and Technology Austria, 2025, doi:<a href="https://doi.org/10.15479/AT-ISTA-19540">10.15479/AT-ISTA-19540</a>. Henheik, S. J. (2025). <i>Modeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-19540">https://doi.org/10.15479/AT-ISTA-19540</a> Henheik SJ. Modeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems. 2025. doi:<a href="https://doi.org/10.15479/AT-ISTA-19540">10.15479/AT-ISTA-19540</a> Henheik SJ. 2025. Modeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems. Institute of Science and Technology Austria. Henheik, Sven Joscha. “Modeling Complex Quantum Systems: Random Matrices, BCS Theory, and Quantum Lattice Systems.” Institute of Science and Technology Austria, 2025. <a href="https://doi.org/10.15479/AT-ISTA-19540">https://doi.org/10.15479/AT-ISTA-19540</a>. S. J. Henheik, “Modeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems,” Institute of Science and Technology Austria, 2025. 195402025-04-10T21:21:18Z2026-07-29T13:18:17Z