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Jakob Möller

Publications and source records attributed to Jakob Möller.

9 recordsLinked to original sources
arXiv2026

Velocity Averaging for the Wigner Equation with Electromagnetic Fields

We discuss the application of $L^2$-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the $L^2$-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables $(x,v=ξ-A(t,x))$, which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in $L^2$ but no gain in regularity.

math.AP↗
arXiv2026

Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime

This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110--125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order $\hbar^{d/2}$, where $d$ is the space dimension and $\hbar$ the reduced Planck constant. In space dimension $d=1$, we prove that the density function belongs to the Sobolev space $H^s(\mathbb R)$ for some $s>0$. In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung's system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above.

math.AP↗
arXiv2025

The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations

We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. Gérard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform $L^2$ bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data.

math.AP↗
arXiv2025

The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods

The self-consistent Pauli-Poisswell and Pauli-Darwin equations for 2-spinors are $O(1/c)$ (where $c$ denotes the speed of light) semi-relativistic approximations of the Dirac-Maxwell equation for 4-spinors coupled to the self-consistent electromagnetic fields generated by the charge and current densities of a fast moving electric charge. They consist of a vector-valued magnetic Schrödinger equation with the Stern-Gerlach term which couples spin and magnetic field, coupled to 1+3 Poisson equations as the magnetostatic approximation of Maxwell's equations. The Pauli-Poisswell and Pauli-Dariwn euqations are $O(1/c)$ models keeping both relativistic effects magnetism and spin, both of which are absent in the non-relativistic Schrödinger-Poisson equation and inconsistent in the magnetic Schrödinger-Maxwell equation. We prove the local in time semiclassical limit $\hbar \rightarrow 0$ to the Euler-Poisswell equation and Euler-Darwin equations based on WKB analysis and energy estimates. Moreover we obtain weak convergence of the monokinetic Wigner transform to the monokinetic scalar Wigner measure solving the Vlasov-Poisswell and Vlasov-Darwin equations and strong convergence of the macroscopic densities. We introduce the Euler-Poisswell/Darwin equation and prove local wellposedness and a blow up alternative.

math.AP↗
arXiv2025

Local Wellposedness and Global Weak Solutions of the Pauli-Darwin/Poisswell Equations

We construct local (in time) strong solutions in {$H^s(\mathbb{R}^3)$, $s>3/2$} and global weak solutions with finite energy for both the Pauli-Darwin and the Pauli-Poisswell systems. These are the first rigorous results on local and global wellposedness for these nonlinear first-order semi-relativistic quantum models for fast moving electrons. The Pauli equation is essentially a vector-valued magnetic Schrödinger equation for a 2-spinor with an additional Stern-Gerlach term coupling spin and magnetic field, keeping terms up to first order in $1/c$, where $c$ denotes the speed of light. The self-consistent electromagnetic field is computed from the charge density and current density by semi-relativistic approximations of the Maxwell equations: the Poisswell equation at $O(1/c)$ and the Darwin equation at $O(1/c^2)$.\\ We present the physics and asymptotic relations and provide proofs that rely on energy estimates for the strong solutions and compactness with an appropriate regularization for the weak solutions.

math.AP↗
arXiv2025

Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space $(x,v)$ containing a transport ("advection") term. By integrating over $v$ in velocity space $\mathbb{R}_v^d$ (velocity averaging), one gains regularity for the density in position space $\mathbb{R}_x^d$. The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in "classical kinetic theory". In "quantum kinetic theory", the Schrödinger equation for the complex-valued "wave function" in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ("Quantum Vlasov") equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the "quantum" case of the Wigner equation. The hard part are the "semi-classical" averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context "pure vs. mixed states" play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the "semi-classical" case which is worked out in full detail in a follow-up article.

math.AP↗
arXiv2023

The Pauli-Poisson equation and its semiclassical limit

The Pauli-Poisson equation is a semi-relativistic model for charged spin-1/2-particles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schrödinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global wellposedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schrödinger-Poisson equation.

math.AP↗
arXiv2023

Nonlinear PDE models in semi-relativistic quantum physics

We present the self-consistent Pauli equation, a semi-relativistic model for charged spin-$1/2$-particles with self-interaction with the electromagnetic field. The Pauli equation arises as the $O(1/c)$ approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac-Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting the correct self-consistent equation is the Schrödinger-Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger-Poisson equation also arises as the mean field limit of the $N$-body Schrödinger equation with Coulomb interaction. We propose that the Pauli-Poisson equation arises as the mean field limit $N \rightarrow \infty$ of the linear $N$-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli-Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in $1/c$ of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb-Thirring estimates.

math-ph↗