My reseach focuses on mathematical questions that have their origin in physcis. The main areas I have worked in are spectral theory and kinetic theory/quantum chaos. My contribution to these areas are explained further below.
Sharp spectral asymptotics in low regularity setting.
The main focus of my research in this direction has been to obtain sharp spectral asymptotics for Schrödinger operators with least possible regularity of the potential. I have published the following three papers in this research direction.
S. Mikkelsen. Sharp semiclassical spectral asymptotics for Schrödinger operators with non-smooth potentials. J. Spectr. Theory 15 (2025), no. 2, pp. 819–846. Preprint Article.
S. Mikkelsen. Sharp semiclassical spectral asymptotics for local magnetic Schrödinger operators on without full regularity. Annales Henri Poincare, 26(5):1865–1906, 2025. Preprint Article.
S. Mikkelsen. Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients. J. Pseudo-Differ. Oper. Appl., 15(1):Paper No. 8, 113, 2024. Preprint Article.
The three papers build on each other, starting with [3], continuing with [2] and culminating in [1].
Application of sharp spectral asymptotics.
Sharp spectral asymptotics have applications to establish different results in mathematical physics. One such application concerns commutator estimates, which were introduced as an assumption in a study of the dynamics for fermions. The first work establishing such estimates for large classes of potentials was the following paper.
Pseudodifferential and functional calculus on nilpotent Lie groups.
Recently, pseudodifferential calculi have been developed for nilpotent Lie groups. Together with a collaborator I have been part of this development by establishing that one of these pseudodifferential calculi also contains the functional calculus. This was done in the following paper.
As an application, this work establishes leading-order Weyl asymptotics for a large class of subelliptic operators.
Kinetic theory and derivation of the linear Boltzmann equation.
One of the fundamental questions in physics is to derive macroscopic transport equations from the underlying microscopic dynamics. One such case is the derivation of the linear Boltzmann equation, which I have obtained in the following paper.
S. Mikkelsen. Schrödinger evolution in a low-density random potential: annealed convergence to the linear Boltzmann equation for general semiclassical Wigner measures. arXiv arXiv:2303.05176, 2023. Link
The proof involves a number of different techniques (Duhamel expansions and their convergence, pseudodifferential theory, Wigner measures, probability theory,…) which makes it very technical.
Quantum chaos.
During my time as a postdoc in Helsinki, I have been working on questions in quantum chaos. One of these has been equidistribution of eigenfunctions in the large genus limit for eigenfunctions of Schrödinger operators. This work has resulted in the following paper.
K. Hippi, F. Lequen, S. Mikkelsen, T. Sahlsten and H. Ueberschär. Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit. arXiv, arXiv:2604.21582, 2026. Link.
This is the first result of this kind for operators that are not homogeneous.
A second project was focused on properties of eigenfunctions in configuration space of rational polygons. Here I together with a collaborator established mixing for eigenfunctions in configuration space for large classes of rational polygons. Moreover we also established this result for arbitrary tori using simple arguments from the theory of Fourier series. These results are contained in the upcomming paper.
K. Hippi, and S. Mikkelsen. Quantum mixing for eigenfunctions of rational polygons in configuration space. arXiv, arXiv:????.?????, 2026.
Theses.
I have made both a PhD. thesis and a master thesis. The content of the PhD. thesis is what turned into the two publications [3] and [4]. In my master's thesis, I studied a recent preprint whose proof relied on a central lemma that was later found to be incorrect. I developed an alternative approach that bypassed this lemma and yielded a new proof of the main theorem.
S. Mikkelsen. Semiclassical Analysis – Optimal Weyl law with an application. PhD. Thesis, Aarhus University 2020. Link.
S. Mikkelsen. A dynamical version of the Bercovici-Pata bijection – Obtained through Hermitian Lévy processes. Master Thesis, Aarhus University 2016.
The banner picture is taken at Little Solsbury Hill overlooking Bath.