arXiv subjects

Søren Mikkelsen

Publications and source records attributed to Søren Mikkelsen.

8 recordsLinked to original sources

arXiv2026

Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit

Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of compact hyperbolic surfaces which is uniformly discrete and Benjamini-Schramm converges to $\mathbb{H}$ and let $\{V_n\}_{n\in\mathbb{N}}$ be a sequence of potentials such that the $L^2$-norm of $V_n$ is $o(1)$ with respect to the volume of $X_n$. We prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$ for bounded observables which are polynomial mixing with respect to the geodesic flow. Examples of such sequences of potentials include point-cloud potentials just below the thermodynamic limit, Hartree potentials for dilute Bose gases in hyperbolic space, and sequences induced by bounded potentials in $L^p(\mathbb{H})$ for some $p>0$. This is the first result of this kind beyond the locally symmetric setting. The proof combines classical geodesic flow mixing on $T^1X_n$, replacing Nevo's ergodic theorem, with the Duhamel formula and recently developed geometric wave-kernel estimates by the first author.

math.SP↗

arXiv2026

Quantum mixing for eigenfunctions of rational polygons in configuration space

The Shnirelman-Zelditch-Colin de Verdière theorem and its weak mixing extension relate quantum ergodicity and quantum mixing with ergodicity and weak mixing of the geodesic flow. Integrable systems, such as the flat torus, do not satisfy either in general. Restricting to position-dependent observables, Marklof and Rudnick established equidistribution for almost all eigenfunctions of rational polygons. In this note, we extend this result to off-diagonal elements for a subset of rational polygons not including the torus using weak mixing of the directional billiard flow for almost all directions. Additionally, we provide a different proof that establishes the same result for $2$-tori.

math.SP↗

arXiv2024

Semiclassical functional calculus on nilpotent Lie groups and their compact nilmanifolds

In this paper, we show that the semiclassical calculus recently developed on nilpotent Lie groups and nilmanifolds include the functional calculus of suitable subelliptic operators. Moreover, we obtain Weyl laws for these operators. Amongst these operators are sub-Laplacians in horizontal divergence form perturbed with a potential and their generalisations.

math.AP↗

arXiv2024

Sharp semiclassical spectral asymptotics for local magnetic Schrödinger operators on $\mathbb{R}^d$ without full regularity

We consider operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$ that locally behave as a magnetic Schrödinger operator. For the magnetic Schrödinger operators we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are Hölder continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.

math.SP↗

arXiv2024

Sharp semiclassical spectral asymptotics for Schrödinger operators with non-smooth potentials

We consider semiclassical Schrödinger operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$. For these operators we establish a sharp spectral asymptotics without full regularity. For the counting function we assume the potential is locally integrable and that the negative part of the potential minus a constant is one time differentiable and the derivative is Hölder continues with parameter $μ\geq1/2$. Moreover we also consider sharp Riesz means of order $γ$ with $γ\in(0,1]$. Here we assume the potential is locally integrable and that the negative part of the potential minus a constant is two time differentiable and the second derivative is Hölder continues with parameter $μ$ that depends on $γ$.

math.SP↗

arXiv2023

Schrödinger evolution in a low-density random potential: annealed convergence to the linear Boltzmann equation for general semiclassical Wigner measures

We consider solutions of the time-dependent Schrödinger equation for a potential localised at the points of a Poisson process. We prove convergence of the phase-space distribution in the annealed Boltzmann-Grad limit to a semiclassical Wigner (or defect) measure and show that it is a solution of the linear Boltzmann equation. Our results hold for a large class of square-integrable initial data associated to Wigner measures, including Langragian states, WKB states and coherent states. This extends important previous work by Eng and Erdős.

math-ph↗

arXiv2022

Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and obtaining optimal spectral asymptotics again under certain regularity conditions. For the Weyl law we assume the coefficients are differentiable with Hölder continuous derivatives and for the Riesz means we assume the coefficients are two times differentiable with Hölder continuous derivatives.

math.SP↗

arXiv2019

An optimal semiclassical bound on certain commutators

We prove an optimal semiclassical bound on the trace norm of the following commutators $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),x]$, $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),-i\hbar\nabla]$ and $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),e^{itx}]$, where $H_\hbar$ is a Schrödinger operator with a semiclassical parameter $\hbar$, $x$ is the position operator and $-i\hbar\nabla$ is the momentum operator. These bounds corresponds to a mean-field version of bounds introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of a fermionic system.

math-ph↗