---
_id: '8603'
abstract:
- lang: eng
  text: We consider the Fröhlich polaron model in the strong coupling limit. It is
    well‐known that to leading order the ground state energy is given by the (classical)
    Pekar energy. In this work, we establish the subleading correction, describing
    quantum fluctuation about the classical limit. Our proof applies to a model of
    a confined polaron, where both the electron and the polarization field are restricted
    to a set of finite volume, with linear size determined by the natural length scale
    of the Pekar problem.
acknowledgement: Partial support through National Science Foundation GrantDMS-1363432
  (R.L.F.) and the European Research Council (ERC) under the Euro-pean Union’s Horizon
  2020 research and innovation programme (grant agreementNo 694227; R.S.), is acknowledged.
  Open access funding enabled and organizedby Projekt DEAL.
article_processing_charge: No
article_type: original
author:
- first_name: Rupert
  full_name: Frank, Rupert
  last_name: Frank
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Frank R, Seiringer R. Quantum corrections to the Pekar asymptotics of a strongly
    coupled polaron. <i>Communications on Pure and Applied Mathematics</i>. 2021;74(3):544-588.
    doi:<a href="https://doi.org/10.1002/cpa.21944">10.1002/cpa.21944</a>
  apa: Frank, R., &#38; Seiringer, R. (2021). Quantum corrections to the Pekar asymptotics
    of a strongly coupled polaron. <i>Communications on Pure and Applied Mathematics</i>.
    Wiley. <a href="https://doi.org/10.1002/cpa.21944">https://doi.org/10.1002/cpa.21944</a>
  chicago: Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar
    Asymptotics of a Strongly Coupled Polaron.” <i>Communications on Pure and Applied
    Mathematics</i>. Wiley, 2021. <a href="https://doi.org/10.1002/cpa.21944">https://doi.org/10.1002/cpa.21944</a>.
  ieee: R. Frank and R. Seiringer, “Quantum corrections to the Pekar asymptotics of
    a strongly coupled polaron,” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 74, no. 3. Wiley, pp. 544–588, 2021.
  ista: Frank R, Seiringer R. 2021. Quantum corrections to the Pekar asymptotics of
    a strongly coupled polaron. Communications on Pure and Applied Mathematics. 74(3),
    544–588.
  mla: Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar Asymptotics
    of a Strongly Coupled Polaron.” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 74, no. 3, Wiley, 2021, pp. 544–88, doi:<a href="https://doi.org/10.1002/cpa.21944">10.1002/cpa.21944</a>.
  short: R. Frank, R. Seiringer, Communications on Pure and Applied Mathematics 74
    (2021) 544–588.
date_created: 2020-10-04T22:01:37Z
date_published: 2021-03-01T00:00:00Z
date_updated: 2023-08-04T11:02:16Z
day: '01'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1002/cpa.21944
ec_funded: 1
external_id:
  isi:
  - '000572991500001'
file:
- access_level: open_access
  checksum: 5f665ffa6e6dd958aec5c3040cbcfa84
  content_type: application/pdf
  creator: dernst
  date_created: 2021-03-11T10:03:30Z
  date_updated: 2021-03-11T10:03:30Z
  file_id: '9236'
  file_name: 2021_CommPureApplMath_Frank.pdf
  file_size: 334987
  relation: main_file
  success: 1
file_date_updated: 2021-03-11T10:03:30Z
has_accepted_license: '1'
intvolume: '        74'
isi: 1
issue: '3'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
page: 544-588
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication: Communications on Pure and Applied Mathematics
publication_identifier:
  eissn:
  - '10970312'
  issn:
  - '00103640'
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quantum corrections to the Pekar asymptotics of a strongly coupled polaron
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 74
year: '2021'
...
---
_id: '721'
abstract:
- lang: eng
  text: 'Let S be a positivity-preserving symmetric linear operator acting on bounded
    functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex
    upper half-plane ℍ has a unique solution m with values in ℍ. We show that the
    z-dependence of this solution can be represented as the Stieltjes transforms of
    a family of probability measures v on ℝ. Under suitable conditions on S, we show
    that v has a real analytic density apart from finitely many algebraic singularities
    of degree at most 3. Our motivation comes from large random matrices. The solution
    m determines the density of eigenvalues of two prominent matrix ensembles: (i)
    matrices with centered independent entries whose variances are given by S and
    (ii) matrices with correlated entries with a translation-invariant correlation
    structure. Our analysis shows that the limiting eigenvalue density has only square
    root singularities or cubic root cusps; no other singularities occur.'
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector
    equations on the complex upper half plane. <i>Communications on Pure and Applied
    Mathematics</i>. 2017;70(9):1672-1705. doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>
  apa: Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions
    to quadratic vector equations on the complex upper half plane. <i>Communications
    on Pure and Applied Mathematics</i>. Wiley-Blackwell. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>
  chicago: Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of
    Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications
    on Pure and Applied Mathematics</i>. Wiley-Blackwell, 2017. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>.
  ieee: O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic
    vector equations on the complex upper half plane,” <i>Communications on Pure and
    Applied Mathematics</i>, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017.
  ista: Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic
    vector equations on the complex upper half plane. Communications on Pure and Applied
    Mathematics. 70(9), 1672–1705.
  mla: Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations
    on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>.
  short: O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics
    70 (2017) 1672–1705.
date_created: 2018-12-11T11:48:08Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2021-01-12T08:12:24Z
day: '01'
department:
- _id: LaEr
doi: 10.1002/cpa.21639
ec_funded: 1
intvolume: '        70'
issue: '9'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1512.03703
month: '09'
oa: 1
oa_version: Submitted Version
page: 1672 - 1705
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Communications on Pure and Applied Mathematics
publication_identifier:
  issn:
  - '00103640'
publication_status: published
publisher: Wiley-Blackwell
publist_id: '6959'
quality_controlled: '1'
scopus_import: 1
status: public
title: Singularities of solutions to quadratic vector equations on the complex upper
  half plane
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2017'
...