Computational Complexity in Quantum Mechanics

ended

The goal of this Fellowship is to derive quantitative estimates on the computational complexity of spectral problems in quantum mechanics. The theoretical framework for this task is provided by the so-called Solvability Complexity Index, which roughly speaking, is the number of successive limits needed to solve the computational problem. I will approach this task by combining techniques from numerical analysis with modern methods from spectral approximation theory. The project is divided into three concise work projects: WP1: NONRELATIVISTIC QUANTUM SYSTEMS. In this project, the spectral problem for Schrödinger operators with various types of potentials is studied. New sharp estimates on the computational complexity are derived. This will contribute to a comprehensive understanding of the nonrelativistic theory. WP2: RESONANCES. In this second project, complexity issues are considered for the computation of scattering resonances in quantum mechanics. I will introduce new mathematical tools, which have not been used in complexity theory before to construct algorithms which compute the set of resonances of Schrödinger operators in one limit. WP3: EXTENSION TO RELATIVISTIC THEORY. The purpose of the final project is to extend the above results to the relativistic setting, in which the Schrödinger operator is replaced by a Dirac operator. This task is far from trivial, as methods from the Schrödinger case are generally not useful for Dirac operators. I also have robust career development and public outreach agendas, to complement the scientific aspects of this proposal. Combined, all these elements will establish me as a prominent research leader upon my return to Germany, with extensive links throughout Europe and the US.