---
_id: '11473'
abstract:
- lang: eng
text: "The polaron model is a basic model of quantum field theory describing a single
particle\r\ninteracting with a bosonic field. It arises in many physical contexts.
We are mostly concerned\r\nwith models applicable in the context of an impurity
atom in a Bose-Einstein condensate as\r\nwell as the problem of electrons moving
in polar crystals.\r\nThe model has a simple structure in which the interaction
of the particle with the field is given\r\nby a term linear in the field’s creation
and annihilation operators. In this work, we investigate\r\nthe properties of
this model by providing rigorous estimates on various energies relevant to the\r\nproblem.
The estimates are obtained, for the most part, by suitable operator techniques
which\r\nconstitute the principal mathematical substance of the thesis.\r\nThe
first application of these techniques is to derive the polaron model rigorously
from first\r\nprinciples, i.e., from a full microscopic quantum-mechanical many-body
problem involving an\r\nimpurity in an otherwise homogeneous system. We accomplish
this for the N + 1 Bose gas\r\nin the mean-field regime by showing that a suitable
polaron-type Hamiltonian arises at weak\r\ninteractions as a low-energy effective
theory for this problem.\r\nIn the second part, we investigate rigorously the
ground state of the model at fixed momentum\r\nand for large values of the coupling
constant. Qualitatively, the system is expected to display\r\na transition from
the quasi-particle behavior at small momenta, where the dispersion relation\r\nis
parabolic and the particle moves through the medium dragging along a cloud of
phonons, to\r\nthe radiative behavior at larger momenta where the polaron decelerates
and emits free phonons.\r\nAt the same time, in the strong coupling regime, the
bosonic field is expected to behave purely\r\nclassically. Accordingly, the effective
mass of the polaron at strong coupling is conjectured to\r\nbe asymptotically
equal to the one obtained from the semiclassical counterpart of the problem,\r\nfirst
studied by Landau and Pekar in the 1940s. For polaron models with regularized
form\r\nfactors and phonon dispersion relations of superfluid type, i.e., bounded
below by a linear\r\nfunction of the wavenumbers for all phonon momenta as in
the interacting Bose gas, we prove\r\nthat for a large window of momenta below
the radiation threshold, the energy-momentum\r\nrelation at strong coupling is
indeed essentially a parabola with semi-latus rectum equal to the\r\nLandau–Pekar
effective mass, as expected.\r\nFor the Fröhlich polaron describing electrons
in polar crystals where the dispersion relation is\r\nof the optical type and
the form factor is formally UV–singular due to the nature of the point\r\ncharge-dipole
interaction, we are able to give the corresponding upper bound. In contrast to\r\nthe
regular case, this requires the inclusion of the quantum fluctuations of the phonon
field,\r\nwhich makes the problem considerably more difficult.\r\nThe results
are supplemented by studies on the absolute ground-state energy at strong coupling,\r\na
proof of the divergence of the effective mass with the coupling constant for a
wide class of\r\npolaron models, as well as the discussion of the apparent UV
singularity of the Fröhlich model\r\nand the application of the techniques used
for its removal for the energy estimates.\r\n"
acknowledged_ssus:
- _id: SSU
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Krzysztof
full_name: Mysliwy, Krzysztof
id: 316457FC-F248-11E8-B48F-1D18A9856A87
last_name: Mysliwy
citation:
ama: 'Mysliwy K. Polarons in Bose gases and polar crystals: Some rigorous energy
estimates. 2022. doi:10.15479/at:ista:11473'
apa: 'Mysliwy, K. (2022). *Polarons in Bose gases and polar crystals: Some rigorous
energy estimates*. Institute of Science and Technology Austria (ISTA). https://doi.org/10.15479/at:ista:11473'
chicago: 'Mysliwy, Krzysztof. “Polarons in Bose Gases and Polar Crystals: Some Rigorous
Energy Estimates.” Institute of Science and Technology Austria (ISTA), 2022. https://doi.org/10.15479/at:ista:11473.'
ieee: 'K. Mysliwy, “Polarons in Bose gases and polar crystals: Some rigorous energy
estimates,” Institute of Science and Technology Austria (ISTA), 2022.'
ista: 'Mysliwy K. 2022. Polarons in Bose gases and polar crystals: Some rigorous
energy estimates. Institute of Science and Technology Austria (ISTA).'
mla: 'Mysliwy, Krzysztof. *Polarons in Bose Gases and Polar Crystals: Some Rigorous
Energy Estimates*. Institute of Science and Technology Austria (ISTA), 2022,
doi:10.15479/at:ista:11473.'
short: 'K. Mysliwy, Polarons in Bose Gases and Polar Crystals: Some Rigorous Energy
Estimates, Institute of Science and Technology Austria (ISTA), 2022.'
date_created: 2022-06-30T12:15:03Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2022-07-06T11:14:02Z
day: '01'
ddc:
- '515'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
doi: 10.15479/at:ista:11473
ec_funded: 1
file:
- access_level: open_access
checksum: 7970714a20a6052f75fb27a6c3e9976e
content_type: application/pdf
creator: kmysliwy
date_created: 2022-07-05T08:12:56Z
date_updated: 2022-07-05T08:12:56Z
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content_type: application/zip
creator: kmysliwy
date_created: 2022-07-05T08:15:52Z
date_updated: 2022-07-05T08:17:12Z
file_id: '11487'
file_name: thes_source.zip
file_size: 5831060
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file_date_updated: 2022-07-05T08:17:12Z
has_accepted_license: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: '138'
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria (ISTA)
related_material:
record:
- id: '8705'
relation: part_of_dissertation
status: public
- id: '10564'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
title: 'Polarons in Bose gases and polar crystals: Some rigorous energy estimates'
type: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2022'
...