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Marcel Griesemer

Publications and source records attributed to Marcel Griesemer.

At least 19 recordsLinked to original sources
arXiv2025

On Rayleigh scattering in the massless Nelson model

Asymptotic completeness of Rayleigh scattering in models of atoms and molecules of non-relativistic QED is expected, but for a proof we still lack sufficient control on the number of emitted soft photons. So far, this obstacle has only been overcome for the spin-boson model. In a general class of models asymptotic completeness holds provided the expectation value of the photon number $N$ remains bounded uniformly in time. This has been established by Faupin and Sigal. We review and simplify their work, and, more importantly, we replace the bound on $N$ by a weaker assumption on the distribution of $N$ that is both necessary and sufficient for asymptotic completeness.

math-ph↗
arXiv2022

On the Weakness of Short-Range Interactions in Fermi Gases

Ultracold quantum gases of equal spin fermions with short range interactions are often considered free even in the presence of strongly binding spin-up-spin-down pairs. We describe a large class of many-particle Schrödinger operators with short-range pair interactions, where this approximation can be justified rigorously.

math-ph↗
arXiv2019

From Short-Range to Contact Interactions in the 1d Bose Gas

For a system of $N$ bosons in one space dimension with two-body $δ$-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.

math-ph↗
arXiv2019

Spectral Theory of the Fermi Polaron

The Fermi polaron refers to a system of free fermions interacting with an impurity particle by means of two-body contact forces. Motivated by the physicists' approach to this system, the present article develops a general mathematical framework for defining many-body Hamiltonians with two-body contact interactions by means of a renormalization procedure. In the case of the Fermi polaron the well-known TMS Hamiltonians are shown to emerge. For the Fermi polaron in a two-dimensional box a novel variational principle, established within the general framework, links the low-lying eigenvalues of the system to the zero-modes of a Birman-Schwinger type operator. It allows us to show, e.g., that the \emph{polaron}- and \emph{molecule} energies, computed in the physical literature, are indeed upper bounds to the ground state energy of the system.

math-ph↗
arXiv2018

Stability of the two-dimensional Fermi polaron

A system composed of an ideal gas of N fermions interacting with an impurity particle in two space dimensions is considered. The interaction between impurity and fermions is given in terms of two-body point interactions whose strength is determined by the two-body binding energy, which is a free parameter of the model. If the mass of the impurity is 1.225 times larger than the mass of a fermion, it is shown that the energy is bounded below uniformly in the number N of fermions. This result improves previous, N-dependent lower bounds and it complements a recent, similar bound for the Fermi polaron in three space dimensions.

math-ph↗
arXiv2017

On the domain of the Nelson Hamiltonian

The Nelson Hamiltonian is unitarily equivalent to a Hamiltonian defined through a closed, semibounded quadratic form, the unitary transformation being explicitly known and due to Gross. In this paper we study mapping properties of the Gross-transform in order to characterize regularity properties of vectors in the form domain of the Nelson Hamiltonian. Since the operator domain is a subset of the form domain, our results apply to vectors in the domain of the Hamiltonian was well. - This work is a continuation of our previous work on the Fröhlich Hamiltonian.

math-ph↗
arXiv2016

Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces

This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive $C_0$-group where the domain $D$ and the growth exponent are independent of $t$. Well-posedness holds provided that $t\mapsto A(t)y$ is Lipschitz for all $y\in D$. Hölder continuity of degree $α<1$ is not sufficient and the assumption of uniform convexity cannot be dropped.

math.AP↗
arXiv2016

On the dynamics of polarons in the strong-coupling limit

The polaron model of H. Fröhlich describes an electron coupled to the quantized longitudinal optical modes of a polar crystal. In the strong-coupling limit one expects that the phonon modes may be treated classically, which leads to a coupled Schrödinger-Poisson system with memory. For the effective dynamics of the electron this amounts to a nonlinear and non-local Schrödinger equation. We use the Dirac-Frenkel variational principle to derive the Schrödinger-Poisson system from the Fröhlich model and we present new results on the accuracy of their solutions for describing the motion of Fröhlich polarons in the strong-coupling limit. Our main result extends to $N$-polaron systems.

math-ph↗
arXiv2016

On the dynamics of the mean-field polaron in the weak-coupling limit

We consider the dynamics of the mean-field polaron in the weak-coupling limit of vanishing electron-phonon interaction, $\varepsilon \to 0$. This is a singular limit formally leading to a Schrödinger--Poisson system that is equivalent to the nonlinear Choquard equation. By establishing estimates between the approximation obtained via the Choquard equation and true solutions of the original system we show that the Choquard equation makes correct predictions about the dynamics of the polaron mean-field model for small values of $\varepsilon > 0$.

math-ph↗
arXiv2014

Kato's theorem on the integration of non-autonomous linear evolution equations

This paper is devoted to a comparison of early works of Kato and Yosida on the integration of non-autonomous linear evolution equations $\dot{x} = A(t)x$ in Banach space, where the domain $D$ of $A(t)$ is independent of $t$. Our focus is on the regularity assumed of $t\mapsto A(t)$ and our main objective is to clarify the meaning of the rather involved set of assumptions given in Yosida's classic and highly influential \emph{Functional Analysis}. We prove Yosida's assumptions to be equivalent to Kato's condition that $t\mapsto A(t)x$ is continuously differentiable for each $x\in D$.

math.FA↗
arXiv2013

The Strong-Coupling Polaron in Electromagnetic Fields

This paper is concerned with Fröhlich polarons subject to external electromagnetic fields in the limit of large electron-phonon coupling. To leading order in the coupling constant, $\sqrtα$, the ground state energy is shown to be correctly given by the minimum of the Pekar functional including the electromagnetic fields, provided these fields in the Fröhlich model are scaled properly with $α$. As a corollary, the binding of two polarons in strong magnetic fields is obtained.

math-ph↗
arXiv2013

Multipolarons in a Constant Magnetic Field

The binding of a system of $N$ polarons subject to a constant magnetic field of strength $B$ is investigated within the Pekar-Tomasevich approximation. In this approximation the energy of $N$ polarons is described in terms of a non-quadratic functional with a quartic term that accounts for the electron-electron self-interaction mediated by phonons. The size of a coupling constant, denoted by $α$, in front of the quartic is determined by the electronic properties of the crystal under consideration, but in any case it is constrained by $0<α<1$. For all values of $N$ and $B$ we find an interval $α_{N,B}<α<1$ where the $N$ polarons bind in a single cluster described by a minimizer of the Pekar-Tomasevich functional. This minimizer is exponentially localized in the $N$-particle configuration space $\R^{3N}$.

math-ph↗
arXiv2011

On the Magnetic Pekar Functional and the Existence of Bipolarons

First, this paper proves the existence of a minimizer for the Pekar functional including a constant magnetic field and possibly some additional local fields that are energy reducing. Second, the existence of the aforementioned minimizer is used to establish the binding of polarons in the model of Pekar-Tomasevich including external fields.

math-ph↗
arXiv2009

On the Atomic Photoeffect in Non-relativistic QED

In this paper we present a mathematical analysis of the photoelectric effect for one-electron atoms in the framework of non-relativistic QED. We treat photo-ionization as a scattering process where in the remote past an atom in its ground state is targeted by one or several photons, while in the distant future the atom is ionized and the electron escapes to spacial infinity. Our main result shows that the ionization probability, to leading order in the fine-structure constant, $α$, is correctly given by formal time-dependent perturbation theory, and, moreover, that the dipole approximation produces an error of only sub-leading order in $α$. In this sense, the dipole approximation is rigorously justified.

math-ph↗
arXiv2009

Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems

For systems of $N$ charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants $U>0$ and $α>0$ measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition $\sqrt{2}α U$ the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and $E_N$ behaves like $-N^{7/3}$.

math-ph↗
arXiv2008

Spectral Renormalization Group

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle in non-relativistic QED near the ground state energy

math-ph↗
arXiv2008

Asymptotic Electromagnetic Fields in Non-relativistic QED: the Problem of Existence Revisited

This paper is devoted to the scattering of photons at electrons in models of non-relativistic quantum mechanical particles coupled minimally to the soft modes of the quantized electromagnetic field. We prove existence of scattering states involving an arbitrary number of asymptotic photons of arbitrarily high energy. Previously, upper bounds on the photon energies seemed necessary in the case of $n>1$ asymptotic photons and non-confined, non-relativistic charged particles.

math-ph↗