arXiv subjects

Jonas Lampart

Publications and source records attributed to Jonas Lampart.

At least 19 recordsLinked to original sources

arXiv2026

Effective Dynamics for the Bose Polaron in the Large-Volume Mean-Field Limit

We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density $\rho$ and large volume $\Lambda$ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint $\Lambda^3 \ll \rho$, the effective description by the translation-invariant Bogoliubov-Fr\"ohlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.

math-ph↗

arXiv2025

A numerical study of stability for solitary waves of a quasi-linear Schr{\"o}dinger equation

We discuss the (in)stability of solitary waves for a quasi-linear Schr{\"o}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the nonlinearity, we determine analytically the asymptotic behavior of the $L^2$-mass of the solution as a function of the frequency close to the critical frequencies, which leads to natural conjectures concerning their stability. Depending on the exponent and the dimension, we expect all solitary waves to be stable, or the emergence of both a stable and an unstable branch of solutions. We investigate our conjectures numerically, and find compatible results both for the mass-energy relation and the dynamics. We observe that perturbations of solitary waves on the unstable branch may converge dynamically to the stable solution of a similar mass, or disperse. More general initial conditions show a similar behavior.

math.AP↗

arXiv2025

Dynamics and equilibrium states of infinite systems of lattice bosons

We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a $C^*$--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.

math-ph↗

arXiv2025

Ultraviolet Renormalization of Spin Boson Models I. Normal and 2-Nilpotent Interactions

We study the ultraviolet problem for models of a finite-dimensional quantum mechanical system linearly coupled to a bosonic quantum field, such as the (many-)spin boson model or its rotating-wave approximation. If the state change of the system upon emission or absorption of a boson is either given by a normal matrix or by a 2-nilpotent one, which is the case for the previously named examples, we prove an optimal renormalization result. We complement it, by proving the norm resolvent convergence of appropriately regularized models to the renormalized one. Our method consists of a dressing transformation argument in the normal case and an appropriate interior boundary condition for the 2-nilpotent case.

math-ph↗

arXiv2025

A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate

In the Bogoliubov-Fröhlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.

math-ph↗

arXiv2024

Equality of magnetization and edge current for interacting lattice fermions at positive temperature

We prove that the magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch's theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.

math-ph↗

arXiv2024

Validity of the Fr\"ohlich model for a mobile impurity in a Bose-Einstein condensate

We analyze the many-body Hamiltonian describing a mobile impurity immersed in a Bose-Einstein condensate (BEC). Using exact unitary transformations and rigorous error estimates, we show the validity of the Bogoliubov-Fr\"ohlich Hamiltonian for the Bose polaron in the regime of moderately strong, repulsive interactions with a dilute BEC. Moreover, we calculate analytically the universal logarithmic correction to the ground state energy that arises from an impurity mediated phonon-phonon interaction.

cond-mat.quant-gas↗

arXiv2024

The excitation spectrum of a Bose gas with an impurity in the Gross-Pitaevskii regime

We study a dilute system of $N$ interacting bosons coupled to an impurity particle via a pair potential in the Gross--Pitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number, and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov--Fr\"ohlich Hamiltonian in the limit $N\to \infty$.

math-ph↗

arXiv2023

Hamiltonians for polaron models with subcritical ultraviolet singularities

We treat the ultraviolet problem for polaron-type models in nonrelativistic quantum field theory. Assuming that the dispersion relations of particles and the field have the same growth at infinity, we cover all subcritical (superrenormalisable) interactions. The Hamiltonian without cutoff is exhibited as an explicit self-adjoint operator obtained by a finite iteration procedure. The cutoff Hamiltonians converge to this operator in the strong resolvent sense after subtraction of a perturbative approximation for the ground-state energy.

math-ph↗

arXiv2023

Renormalized Bogoliubov Theory for the Nelson Model

We consider the time evolution of the renormalized Nelson model, which describes $N$ bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles $N\gg 1$ with coupling constant proportional to $N^{-1/2}$. First, we show that initial states exhibiting Bose-Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger-Klein-Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger-Klein-Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.

math-ph↗

arXiv2022

On the global minimum of the energy-momentum relation for the polaron

For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.

math-ph↗

arXiv2021

The Dirac-Klein-Gordon system in the strong coupling limit

We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac-Fock equation on the space of Hilbert-Schmidt operators.

math.AP↗

arXiv2021

The Nelson Model on Static Spacetimes

The Nelson model describes the interaction of nonrelativistic quantum particles with a relativistic quantum field of scalar bosons. Nelson rigorously demonstrated in 1964 the existence of a well-defined self-adjoint Nelson Hamiltonian by renormalisation. Recently, a Fock space description of the renormalised Nelson Hamiltonian and its domain were found based on a novel approach to defining Hamiltonians in quantum field theories. This novel approach involves an interior boundary condition (IBC) on the state vectors, which relates the value of the wave function on the boundary of the configuration space to the value at a point in the interior. In the present work, we apply the recently developed techniques to the Nelson model on static spacetimes.

math-ph↗

arXiv2021

Dynamics of a tracer particle interacting with excitations of a Bose-Einstein condensate

We consider the quantum dynamics of a large number $N$ of interacting bosons coupled a tracer particle, i.e. a particle of another kind, on a torus. We assume that in the initial state the bosons essentially form a homogeneous Bose-Einstein condensate, with some excitations. With an appropriate mean-field scaling of the interactions, we prove that the effective dynamics for $N\to \infty$ is generated by the Bogoliubov-Fröhlich Hamiltonian, which couples the tracer particle linearly to the excitation field.

math-ph↗

arXiv2021

The resolvent of the Nelson Hamiltonian improves positivity

We prove that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum P improves positivity in the (momentum) Fock-representation, for every P. Our argument is based on an explcit representation of the renormalised operator and its domain using interior boundary conditions.

math.FA↗

arXiv2021

An abstract framework for interior-boundary conditions

In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive perturbations. Such boundary conditions have recently provided insight into problems form quantum field theory. We discuss interior-boundary conditions in an abstract setting, with a focus on self-adjoint operators, proving self-adjointness criteria, resolvent formulas, and a classification theorem.

math.SP↗

arXiv2020

The Renormalised Bogoliubov-Fröhlich Hamiltonian

The Bogoliubov-Fröhlich Hamiltonian models the interaction of an impurity with the excitations of a Bose-Einstein condensate. It has been observed that the dependence of the ground state energy on the ultraviolet cutoff differs significantly from what would be expected from similar well-known models. We give a detailed explanation of this UV behaviour, and provide an explicit representation of the renormalised Hamiltonian.

cond-mat.quant-gas↗

arXiv2020

A remark on the attainable set of the Schrödinger equation

We discuss the set of wavefunctions $ψ_V(t)$ that can be obtained from a given initial condition $ψ_0$ by applying the flow of the Schrödinger operator $-Δ+ V(t,x)$ and varying the potential $V(t,x)$. We show that this set has empty interior, both as a subset of the sphere in $L^2(\mathbb{R}^d)$ and as a set of trajectories.

math-ph↗