# Microscopic Derivation of Time-dependent Point Interactions

@article{Carlone2019MicroscopicDO, title={Microscopic Derivation of Time-dependent Point Interactions}, author={Raffaele Carlone and Michele Correggi and Marco Falconi and Marco Olivieri}, journal={arXiv: Mathematical Physics}, year={2019} }

We study the dynamics of the three-dimensional Frohlich polaron -- a quantum particle coupled to a bosonic field -- in the quasi-classical regime, i.e., when the field is very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As…

## 7 Citations

### Derivation of the Landau–Pekar Equations in a Many-Body Mean-Field Limit

- PhysicsArchive for Rational Mechanics and Analysis
- 2021

The Fröhlich Hamiltonian is considered in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field and it is shown that the dynamics of the system is approximately described by the Landau–Pekar equations.

### Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model

- MathematicsReviews in Mathematical Physics
- 2022

We study the time evolution of the Nelson model in a mean-field limit in which N nonrelativistic bosons weakly couple (w.r.t. the particle number) to a positive or zero mass quantized scalar field.…

### Norm approximation for the Fr\"ohlich dynamics in the mean-field regime

- Physics
- 2022

We study the time evolution of the Fr¨ohlich Hamiltonian in a mean-ﬁeld limit in which many particles weakly couple to the quantized phonon ﬁeld. Assuming that the particles are initially in a…

### Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

- PhysicsPure and Applied Analysis
- 2021

We consider the Frohlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding…

### Quasi-Classical Dynamics

- Physics
- 2019

We study quantum particles in interaction with a force-carrying field, in the quasi-classical limit. This limit is characterized by the field having a very large number of excitations (it is…

### Ground State Properties in the Quasi-Classical Regime

- Physics
- 2020

We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The…

### Complete Ionization for a Non-autonomous Point Interaction Model in d = 2

- MathematicsCommunications in Mathematical Physics
- 2022

We consider the two dimensional Schrödinger equation with a time dependent point interaction, which represents a model for the dynamics of a quantum particle subject to a point interaction whose…

## References

SHOWING 1-10 OF 57 REFERENCES

### Two-dimensional time-dependent point interactions

- Mathematics, Physics
- 2017

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schr\"{o}dinger…

### Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit

- Mathematics, Physics
- 2017

We study the quasi-classical limit of a quantum system composed of finitely many nonrelativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes…

### Ionization for Three Dimensional Time-Dependent Point Interactions

- Mathematics
- 2005

We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength” of the interaction α(t)…

### Schrodinger Equation with Moving Point Interactions in Three Dimensions

- Mathematics
- 1999

We consider the motion of a non relativistic quantum particle in R^3 subject to n point interactions which are moving on given smooth trajectories. Due to the singular character of the time-dependent…

### Rotating Singular Perturbations of the Laplacian

- Mathematics
- 2003

Abstract.
We study a system of a quantum particle interacting with a singular time-dependent
uniformly rotating potential in 2 and 3 dimensions: in particular we
consider an interaction with support…

### The classical field limit of scattering theory for non-relativistic many-boson systems. II

- Physics, Mathematics
- 1979

We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected…

### Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

- Mathematics, PhysicsJournal of Spectral Theory
- 2019

We study the quasi-classical limit of the Pauli-Fierz model: the system is composed of finitely many non-relativistic charged particles interacting with a bosonic radiation field. We trace out the…

### On the dynamics of polarons in the strong-coupling limit

- Physics, Mathematics
- 2016

The polaron model of H. Frohlich 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…

### Partially Classical Limit of the Nelson Model

- Physics
- 2004

Abstract.We consider the Nelson model which describes a quantum system of nonrelativistic identical particles coupled to a possibly massless scalar Bose field through a Yukawa type interaction. We…

### Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

- Mathematics, Physics
- 2007

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the…