In my research, I develop and apply tools from functional analysis, partial differential equations, and semiclassical analysis to study the properties of effective evolution equations and to justify their validity as approximate descriptions of the dynamics of complex many-body quantum systems. 

Effective equations arise because nature behaves in distinct ways on different scales. Phenomenological observations and theoretical reasoning have therefore led to various physical theories that describe phenomena in certain situations. Usually, one theory is regarded as the most precise, with others serving as approximations that are easier to investigate but only applicable in specific contexts. 

For example, light is described as a ray in geometrical optics, as a wave in classical electromagnetism, and as being composed of energy quanta (photons) in quantum electrodynamics. While quantum electrodynamics is the most precise theory, from which the others emerge as approximations valid in specific situations, geometrical optics, for instance, is much easier to use when predicting how a light beam reflects off a mirror. 

In physics, the link between different theories is often obtained through heuristic arguments. A rigorous mathematical study using asymptotic analysis quantifies the error introduced by the approximation and can provide a deeper understanding of the mechanisms that allow for the use of these approximate descriptions. 

The focus of my research is on effective evolution equations for quantum systems, specifically on effective descriptions for non-relativistic quantum field models, which are used in condensed matter physics or as approximate descriptions of quantum electrodynamics. More recently, I have also worked on the derivation of kinetic equations for quantum systems.

In 2018, I obtained my Ph.D. in mathematics from the Ludwig Maximilian University of Munich under the supervision of Peter Pickl.

I spent two years as postdoctoral researcher in Robert Seiringer’s group at the Institute of Science and Technology Austria, followed by five years in Chiara Saffirio’s group at the University of Basel. Since December 2024, I have been at Constructor University.

• Norm approximation for the Fröhlich dynamics in the mean-field regime, N. Leopold, 
  J. Funct. Anal. 285(4), 109979 (2023), DOI: 10.1016/j.jfa.2023.109979.
• Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation,
  N. Leopold and C. Saffirio,
  SIAM J. Math. Anal. 55(3), 1676-1706 (2023), DOI: 10.1137/22M14936. 
• Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron, 
  N. Leopold, D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer,
  Pure Appl. Anal. 3(4), 653-676 (2021), DOI: 10.2140/paa.2021.3.653. 
• The Landau-Pekar equations: Adiabatic theorem and accuracy, 
  N. Leopold, S. Rademacher, B. Schlein and R. Seiringer, 
  Anal. PDE 14(7), 2079-2100 (2021), DOI: 10.2140/apde.2021.14.2079.
• Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian,
  N. Leopold and P. Pickl,
  SIAM J. Math. Anal. 52(5), 4900-4936 (2020), DOI: 10.1137/19M1307639. 
• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions, 
  M. Jeblick, N. Leopold and P. Pickl, Commun. Math. Phys. 372, 1–69 (2019), DOI: 10.1007/s00220-019-03599-x. 
   

Full list of publications