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Ioannis Anapolitanos

Publications and source records attributed to Ioannis Anapolitanos.

11 recordsLinked to original sources
arXiv2026

The Lieb--Thomas strategy for strongly coupled fermionic multipolarons with general external fields

In this article, we prove that the ground-state energy of a fermionic Fröhlich multipolaron can be approximated, in the strong electron-phonon coupling limit, by the ground-state energy of a corresponding fermionic Pekar-Tomasevich multipolaron, even in the presence of external electric and magnetic fields. Our analysis builds upon Lieb and Thomas' approach \cite{liebthomas}, which was originally developed for a single polaron without external fields, and Wellig's generalization to multipolarons \cite{wellig} with (specialized) external fields. Our main new contributions are twofold. First, we take into account the fermionic statistics of the multipolaron by employing a localization method from \cite{liebloss}. Second, we relax an assumption in \cite{wellig} on the external electric and magnetic fields, which is not easily verifiable unless the fields are periodic. Instead, we allow for general fields that only ensure self-adjointness of the Fröhlich Hamiltonian. In particular, our work demonstrates the robustness of the Lieb--Thomas strategy when extended to fermionic multipolarons and general external potentials.

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arXiv2023

On boundedness of isomerization paths for non- and semirelativistic molecules

This article focuses on isomerizations of molecules, i.e. chemical reactions during which a molecule is transformed into another one with the same atoms in a different spatial configuration. We consider the special case in which the system breaks into two submolecules whose internal geometry is solid during the whole procedure. We prove, under some conditions, that the distance between the two submolecules stays bounded during the entire reaction. To this end, we provide an asymptotic expansion of the interaction energy between two molecules, including multipolar interactions and the van der Waals attraction. In addition to this static result, we proceed to a quasistatic analysis to investigate the variation of the energy when the nuclei move. This paper generalizes a recent work by M. Lewin and the first author in two directions. The first one is that we relax the assumption that the ground state eigenspaces of the submolecules have to fulfill. The second one is that we allow semirelativistic kinetic energy as well.

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arXiv2022

On the van der Waals interaction between a molecule and a half-infinite plate

We consider a molecule in the Born-Oppenheimer approximation interacting with a plate of infinite thickness, i.e, a half--space, which is perfectly conducting or dielectric. It is well--known in the physics literature that in this case the atom or molecule is attracted by the plate at sufficiently large distances. This effect is analogous to the well--known van der Waals interaction between neutral atoms or molecules. We prove that the interaction energy $W$ of the system is given by $W(r,v)= -C(v)r^{-3} + \mathcal{O}(r^{-4})$, where $r$ is the distance between the molecule and the plate and $v$ indicates their relative orientation. Moreover, $C(v)$ is positive and continuous, thus the atom or molecule is always pulled towards the plate at sufficiently large distances, for all relative orientations $v$. For some specific systems we provide sharper estimates of $W(r,v)$. This asymptotic behavior is well--known in the physics literature, however, we are not aware of any previous rigorous results, even on existence of a ground state of the system. For pedagogical reasons, we often start with the case of a hydrogen atom and then we generalize the arguments to deal with a general molecule.

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arXiv2019

Compactness of molecular reaction paths in quantum mechanics

We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schrödinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes over all possible paths the highest value of the energy along the paths. Here we state a conjecture about the compactness of min-maxing sequences of such paths, which we then partly solve in the particular case of a molecule composed of two rigid sub-molecules that can move freely in space. More precisely, under appropriate assumptions on the multipoles of the two molecules, we are able to prove that the distance between them stays bounded during the whole chemical reaction. We obtain a critical point at the mountain pass level, which is called a transition state in chemistry. Our method requires to study the critical points and the Morse indices of the classical multipole interactions, as well as to improve existing results about the van der Waals force. This paper generalizes previous works by the second author in several directions.

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arXiv2019

Differentiability of the van der Waals interaction between two atoms

In this work we improve upon previous results on the expansion of the interaction energy of two atoms. On the one hand we prove the van der Waals-London's law, assuming that only one of the ground state eigenspaces of the atoms is irreducible in an appropriate sense. On the other hand we prove strict monotonicity of the interaction energy at large distances and, under more restrictive assumptions, we provide the leading order of its first two derivatives. The first derivative is interpreted as the force in Physics. Moreover, the estimates of the first two derivatives provide a rigorous proof of the monotonicity and concavity of the interaction energy at large distances.

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arXiv2017

Derivation of the Hartree equation for compound Bose gases in the mean field limit

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field $A\in L^2_{\text{loc}}(\R^3;\R^3)$ and the semi-relativistic case.

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arXiv2016

A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the k-th marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see \cite{MS}, \cite{Lu}. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H^1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional $a_N$ introduced by Pickl in \cite{Pi}.

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arXiv2014

Remainder estimates for the Long Range Behavior of the van der Waals interaction energy

The van der Waals-London's law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring [LT], by proving an upper bound which confirms this law. Recently the van der Waals-London's law was proven under some assumptions by I.M. Sigal and the author [AS]. Following the strategy of [AS] and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, using an appropriate test function, we prove an upper bound for the interaction energy, which is sharp to leading order. For the upper bound, our assumptions are weaker, the remainder estimates stronger and the proof is simpler. The upper bound, for the cases it applies, improves considerably the upper bound of Lieb and Thirring. However, their bound is much more general. Here we consider only spinless Fermions.

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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 $\R^{3N}$.

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arXiv2012

The ground state energy of the multi-polaron in the strong coupling limit

We consider the Fröhlich $N$-polaron Hamiltonian in the strong coupling limit and bound the ground state energy from below. In particular, our lower bound confirms that the ground state energy of the Fröhlich polaron and the ground state energy of the associated Pekar-Tomasevich variational problem are asymptotically equal in the strong coupling limit. We generalize the operator approach that was used to prove a similar result in the N=1 case in Lieb and Thomas (1997) and apply a Feynman-Kac formula to obtain the same result for an arbitrary particle number $N \geq 1$.

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