---
OA_place: repository
OA_type: green
_id: '22021'
abstract:
- lang: eng
  text: "We establish global well-posedness for both the defocusing and\r\nfocusing
    complex-valued modified Korteweg–de Vries equations on the real line\r\nin modulation
    spaces Ms,2p (R), for all 1 \x14 p < 1 and 0 \x14 s < 3/2 − 1/p. We\r\nwill also
    show that such solutions admit global-in-time bounds in these spaces\r\nand that
    equicontinuous sets of initial data lead to equicontinuous ensembles\r\nof orbits.
    Indeed, such information forms a crucial part of our well-posedness\r\nargument."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Saikatul
  full_name: Haque, Saikatul
  last_name: Haque
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Yunfeng
  full_name: Zhang, Yunfeng
  last_name: Zhang
citation:
  ama: Haque S, Killip R, Vişan M, Zhang Y.  Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces. <i>Pure and Applied
    Analysis</i>. 2025;7(3):615-637. doi:<a href="https://doi.org/10.2140/paa.2025.7.615">10.2140/paa.2025.7.615</a>
  apa: Haque, S., Killip, R., Vişan, M., &#38; Zhang, Y. (2025).  Global well-posedness
    and equicontinuity for modified Korteweg–de Vries equations in modulation spaces.
    <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/paa.2025.7.615">https://doi.org/10.2140/paa.2025.7.615</a>
  chicago: Haque, Saikatul, Rowan Killip, Monica Vişan, and Yunfeng Zhang. “ Global
    Well-Posedness and Equicontinuity for Modified Korteweg–de Vries Equations in
    Modulation Spaces.” <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers,
    2025. <a href="https://doi.org/10.2140/paa.2025.7.615">https://doi.org/10.2140/paa.2025.7.615</a>.
  ieee: S. Haque, R. Killip, M. Vişan, and Y. Zhang, “ Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces,” <i>Pure and Applied
    Analysis</i>, vol. 7, no. 3. Mathematical Sciences Publishers, pp. 615–637, 2025.
  ista: Haque S, Killip R, Vişan M, Zhang Y. 2025.  Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces. Pure and Applied
    Analysis. 7(3), 615–637.
  mla: Haque, Saikatul, et al. “ Global Well-Posedness and Equicontinuity for Modified
    Korteweg–de Vries Equations in Modulation Spaces.” <i>Pure and Applied Analysis</i>,
    vol. 7, no. 3, Mathematical Sciences Publishers, 2025, pp. 615–37, doi:<a href="https://doi.org/10.2140/paa.2025.7.615">10.2140/paa.2025.7.615</a>.
  short: S. Haque, R. Killip, M. Vişan, Y. Zhang, Pure and Applied Analysis 7 (2025)
    615–637.
das_tickbox: '1'
date_created: 2026-06-19T07:30:23Z
date_published: 2025-06-18T00:00:00Z
date_updated: 2026-06-24T13:22:40Z
day: '18'
doi: 10.2140/paa.2025.7.615
extern: '1'
external_id:
  arxiv:
  - '2411.05300'
fulldoi: https://doi.org/10.2140/paa.2025.7.615
intvolume: '         7'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2411.05300
mathsc:
- 35Q53
- 35Q55
- 37K10
month: '06'
oa: 1
oa_version: Preprint
page: 615-637
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: ' Global well-posedness and equicontinuity for modified Korteweg–de Vries equations
  in modulation spaces'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '22027'
abstract:
- lang: eng
  text: We demonstrate that in three space dimensions, the scattering behaviour of
    semilinear wave equations with quintic-type nonlinearities uniquely determines
    the nonlinearity. The nonlinearity is permitted to depend on both space and time.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nicholas
  full_name: Hu, Nicholas
  last_name: Hu
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Hu N, Killip R, Vişan M. Deconvolutional determination of the nonlinearity
    in a semilinear wave equation. <i>Pure and Applied Analysis</i>. 2025;7(1):1-17.
    doi:<a href="https://doi.org/10.2140/paa.2025.7.1">10.2140/paa.2025.7.1</a>
  apa: Hu, N., Killip, R., &#38; Vişan, M. (2025). Deconvolutional determination of
    the nonlinearity in a semilinear wave equation. <i>Pure and Applied Analysis</i>.
    Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/paa.2025.7.1">https://doi.org/10.2140/paa.2025.7.1</a>
  chicago: Hu, Nicholas, Rowan Killip, and Monica Vişan. “Deconvolutional Determination
    of the Nonlinearity in a Semilinear Wave Equation.” <i>Pure and Applied Analysis</i>.
    Mathematical Sciences Publishers, 2025. <a href="https://doi.org/10.2140/paa.2025.7.1">https://doi.org/10.2140/paa.2025.7.1</a>.
  ieee: N. Hu, R. Killip, and M. Vişan, “Deconvolutional determination of the nonlinearity
    in a semilinear wave equation,” <i>Pure and Applied Analysis</i>, vol. 7, no.
    1. Mathematical Sciences Publishers, pp. 1–17, 2025.
  ista: Hu N, Killip R, Vişan M. 2025. Deconvolutional determination of the nonlinearity
    in a semilinear wave equation. Pure and Applied Analysis. 7(1), 1–17.
  mla: Hu, Nicholas, et al. “Deconvolutional Determination of the Nonlinearity in
    a Semilinear Wave Equation.” <i>Pure and Applied Analysis</i>, vol. 7, no. 1,
    Mathematical Sciences Publishers, 2025, pp. 1–17, doi:<a href="https://doi.org/10.2140/paa.2025.7.1">10.2140/paa.2025.7.1</a>.
  short: N. Hu, R. Killip, M. Vişan, Pure and Applied Analysis 7 (2025) 1–17.
date_created: 2026-06-19T07:36:00Z
date_published: 2025-01-22T00:00:00Z
date_updated: 2026-06-24T13:24:38Z
day: '22'
doi: 10.2140/paa.2025.7.1
extern: '1'
external_id:
  arxiv:
  - '2307.00829'
fulldoi: https://doi.org/10.2140/paa.2025.7.1
intvolume: '         7'
issue: '1'
keyword:
- dispersive equations
- nonlinear wave equation
- semilinear wave equation
- scattering
- inverse scattering
- deconvolution
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2307.00829
mathsc:
- 35L70
- 35P25
- 35R30
month: '01'
oa: 1
oa_version: Preprint
page: 1-17
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Deconvolutional determination of the nonlinearity in a semilinear wave equation
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '14854'
abstract:
- lang: eng
  text: We study the spectrum of the Fröhlich Hamiltonian for the polaron at fixed
    total momentum. We prove the existence of excited eigenvalues between the ground
    state energy and the essential spectrum at strong coupling. In fact, our main
    result shows that the number of excited energy bands diverges in the strong coupling
    limit. To prove this we derive upper bounds for the min-max values of the corresponding
    fiber Hamiltonians and compare them with the bottom of the essential spectrum,
    a lower bound on which was recently obtained by Brooks and Seiringer (Comm. Math.
    Phys. 404:1 (2023), 287–337). The upper bounds are given in terms of the ground
    state energy band shifted by momentum-independent excitation energies determined
    by an effective Hamiltonian of Bogoliubov type.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: David Johannes
  full_name: Mitrouskas, David Johannes
  id: cbddacee-2b11-11eb-a02e-a2e14d04e52d
  last_name: Mitrouskas
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Mitrouskas DJ, Seiringer R. Ubiquity of bound states for the strongly coupled
    polaron. <i>Pure and Applied Analysis</i>. 2023;5(4):973-1008. doi:<a href="https://doi.org/10.2140/paa.2023.5.973">10.2140/paa.2023.5.973</a>
  apa: Mitrouskas, D. J., &#38; Seiringer, R. (2023). Ubiquity of bound states for
    the strongly coupled polaron. <i>Pure and Applied Analysis</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/paa.2023.5.973">https://doi.org/10.2140/paa.2023.5.973</a>
  chicago: Mitrouskas, David Johannes, and Robert Seiringer. “Ubiquity of Bound States
    for the Strongly Coupled Polaron.” <i>Pure and Applied Analysis</i>. Mathematical
    Sciences Publishers, 2023. <a href="https://doi.org/10.2140/paa.2023.5.973">https://doi.org/10.2140/paa.2023.5.973</a>.
  ieee: D. J. Mitrouskas and R. Seiringer, “Ubiquity of bound states for the strongly
    coupled polaron,” <i>Pure and Applied Analysis</i>, vol. 5, no. 4. Mathematical
    Sciences Publishers, pp. 973–1008, 2023.
  ista: Mitrouskas DJ, Seiringer R. 2023. Ubiquity of bound states for the strongly
    coupled polaron. Pure and Applied Analysis. 5(4), 973–1008.
  mla: Mitrouskas, David Johannes, and Robert Seiringer. “Ubiquity of Bound States
    for the Strongly Coupled Polaron.” <i>Pure and Applied Analysis</i>, vol. 5, no.
    4, Mathematical Sciences Publishers, 2023, pp. 973–1008, doi:<a href="https://doi.org/10.2140/paa.2023.5.973">10.2140/paa.2023.5.973</a>.
  short: D.J. Mitrouskas, R. Seiringer, Pure and Applied Analysis 5 (2023) 973–1008.
corr_author: '1'
date_created: 2024-01-22T08:24:23Z
date_published: 2023-12-15T00:00:00Z
date_updated: 2025-04-23T14:21:39Z
day: '15'
department:
- _id: RoSe
doi: 10.2140/paa.2023.5.973
external_id:
  arxiv:
  - '2211.03606'
fulldoi: https://doi.org/10.2140/paa.2023.5.973
intvolume: '         5'
issue: '4'
keyword:
- General Medicine
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2211.03606
month: '12'
oa: 1
oa_version: Preprint
page: 973-1008
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Ubiquity of bound states for the strongly coupled polaron
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 5
year: '2023'
...
---
_id: '14889'
abstract:
- lang: eng
  text: We consider the Fröhlich Hamiltonian with large coupling constant α. For initial
    data of Pekar product form with coherent phonon field and with the electron minimizing
    the corresponding energy, we provide a norm approximation of the evolution, valid
    up to times of order α2. The approximation is given in terms of a Pekar product
    state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics
    taking quantum fluctuations into account. This allows us to show that the Landau-Pekar
    equations approximately describe the evolution of the electron- and one-phonon
    reduced density matrices under the Fröhlich dynamics up to times of order α2.
acknowledgement: "Financial support by the European Union’s Horizon 2020 research
  and innovation programme\r\nunder the Marie Skłodowska-Curie grant agreement No.
  754411 (S.R.) and the European\r\nResearch Council under grant agreement No. 694227
  (N.L. and R.S.), as well as by the SNSF\r\nEccellenza project PCEFP2 181153 (N.L.),
  the NCCR SwissMAP (N.L. and B.S.) and by the\r\nDeutsche Forschungsgemeinschaft
  (DFG) through the Research Training Group 1838: Spectral\r\nTheory and Dynamics
  of Quantum Systems (D.M.) is gratefully acknowledged. B.S. gratefully\r\nacknowledges
  financial support from the Swiss National Science Foundation through the Grant\r\n“Dynamical
  and energetic properties of Bose-Einstein condensates” and from the European\r\nResearch
  Council through the ERC-AdG CLaQS (grant agreement No 834782). D.M. thanks\r\nMarcel
  Griesemer for helpful discussions."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikolai K
  full_name: Leopold, Nikolai K
  id: 4BC40BEC-F248-11E8-B48F-1D18A9856A87
  last_name: Leopold
  orcid: 0000-0002-0495-6822
- first_name: David Johannes
  full_name: Mitrouskas, David Johannes
  id: cbddacee-2b11-11eb-a02e-a2e14d04e52d
  last_name: Mitrouskas
- first_name: Simone Anna Elvira
  full_name: Rademacher, Simone Anna Elvira
  id: 856966FE-A408-11E9-977E-802DE6697425
  last_name: Rademacher
  orcid: 0000-0001-5059-4466
- first_name: Benjamin
  full_name: Schlein, Benjamin
  last_name: Schlein
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Leopold NK, Mitrouskas DJ, Rademacher SAE, Schlein B, Seiringer R. Landau–Pekar
    equations and quantum fluctuations for the dynamics of a strongly coupled polaron.
    <i>Pure and Applied Analysis</i>. 2021;3(4):653-676. doi:<a href="https://doi.org/10.2140/paa.2021.3.653">10.2140/paa.2021.3.653</a>
  apa: Leopold, N. K., Mitrouskas, D. J., Rademacher, S. A. E., Schlein, B., &#38;
    Seiringer, R. (2021). Landau–Pekar equations and quantum fluctuations for the
    dynamics of a strongly coupled polaron. <i>Pure and Applied Analysis</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/paa.2021.3.653">https://doi.org/10.2140/paa.2021.3.653</a>
  chicago: Leopold, Nikolai K, David Johannes Mitrouskas, Simone Anna Elvira Rademacher,
    Benjamin Schlein, and Robert Seiringer. “Landau–Pekar Equations and Quantum Fluctuations
    for the Dynamics of a Strongly Coupled Polaron.” <i>Pure and Applied Analysis</i>.
    Mathematical Sciences Publishers, 2021. <a href="https://doi.org/10.2140/paa.2021.3.653">https://doi.org/10.2140/paa.2021.3.653</a>.
  ieee: N. K. Leopold, D. J. Mitrouskas, S. A. E. Rademacher, B. Schlein, and R. Seiringer,
    “Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly
    coupled polaron,” <i>Pure and Applied Analysis</i>, vol. 3, no. 4. Mathematical
    Sciences Publishers, pp. 653–676, 2021.
  ista: Leopold NK, Mitrouskas DJ, Rademacher SAE, Schlein B, Seiringer R. 2021. Landau–Pekar
    equations and quantum fluctuations for the dynamics of a strongly coupled polaron.
    Pure and Applied Analysis. 3(4), 653–676.
  mla: Leopold, Nikolai K., et al. “Landau–Pekar Equations and Quantum Fluctuations
    for the Dynamics of a Strongly Coupled Polaron.” <i>Pure and Applied Analysis</i>,
    vol. 3, no. 4, Mathematical Sciences Publishers, 2021, pp. 653–76, doi:<a href="https://doi.org/10.2140/paa.2021.3.653">10.2140/paa.2021.3.653</a>.
  short: N.K. Leopold, D.J. Mitrouskas, S.A.E. Rademacher, B. Schlein, R. Seiringer,
    Pure and Applied Analysis 3 (2021) 653–676.
corr_author: '1'
date_created: 2024-01-28T23:01:43Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2025-04-14T07:27:00Z
day: '01'
department:
- _id: RoSe
doi: 10.2140/paa.2021.3.653
ec_funded: 1
external_id:
  arxiv:
  - '2005.02098'
fulldoi: https://doi.org/10.2140/paa.2021.3.653
intvolume: '         3'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2005.02098
month: '10'
oa: 1
oa_version: Preprint
page: 653-676
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly
  coupled polaron
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 3
year: '2021'
...
---
_id: '14890'
abstract:
- lang: eng
  text: We consider a system of N interacting bosons in the mean-field scaling regime
    and construct corrections to the Bogoliubov dynamics that approximate the true
    N-body dynamics in norm to arbitrary precision. The N-independent corrections
    are given in terms of the solutions of the Bogoliubov and Hartree equations and
    satisfy a generalized form of Wick's theorem. We determine the n-point correlation
    functions of the excitations around the condensate, as well as the reduced densities
    of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point
    functions of a quasi-free state and the solution of the Hartree equation. In this
    way, the complex problem of computing all n-point correlation functions for an
    interacting N-body system is essentially reduced to the problem of solving the
    Hartree equation and the PDEs for the Bogoliubov two-point functions.
acknowledgement: "We are grateful for the hospitality of Central China Normal University
  (CCNU),\r\nwhere parts of this work were done, and thank Phan Th`anh Nam, Simone\r\nRademacher,
  Robert Seiringer and Stefan Teufel for helpful discussions. L.B. gratefully acknowledges
  the support by the German Research Foundation (DFG) within the Research\r\nTraining
  Group 1838 “Spectral Theory and Dynamics of Quantum Systems”, and the funding\r\nfrom
  the European Union’s Horizon 2020 research and innovation programme under the Marie\r\nSk
  lodowska-Curie Grant Agreement No. 754411."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Lea
  full_name: Bossmann, Lea
  id: A2E3BCBE-5FCC-11E9-AA4B-76F3E5697425
  last_name: Bossmann
  orcid: 0000-0002-6854-1343
- first_name: Sören P
  full_name: Petrat, Sören P
  id: 40AC02DC-F248-11E8-B48F-1D18A9856A87
  last_name: Petrat
  orcid: 0000-0002-9166-5889
- first_name: Peter
  full_name: Pickl, Peter
  last_name: Pickl
- first_name: Avy
  full_name: Soffer, Avy
  last_name: Soffer
citation:
  ama: Bossmann L, Petrat SP, Pickl P, Soffer A. Beyond Bogoliubov dynamics. <i>Pure
    and Applied Analysis</i>. 2021;3(4):677-726. doi:<a href="https://doi.org/10.2140/paa.2021.3.677">10.2140/paa.2021.3.677</a>
  apa: Bossmann, L., Petrat, S. P., Pickl, P., &#38; Soffer, A. (2021). Beyond Bogoliubov
    dynamics. <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers.
    <a href="https://doi.org/10.2140/paa.2021.3.677">https://doi.org/10.2140/paa.2021.3.677</a>
  chicago: Bossmann, Lea, Sören P Petrat, Peter Pickl, and Avy Soffer. “Beyond Bogoliubov
    Dynamics.” <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers,
    2021. <a href="https://doi.org/10.2140/paa.2021.3.677">https://doi.org/10.2140/paa.2021.3.677</a>.
  ieee: L. Bossmann, S. P. Petrat, P. Pickl, and A. Soffer, “Beyond Bogoliubov dynamics,”
    <i>Pure and Applied Analysis</i>, vol. 3, no. 4. Mathematical Sciences Publishers,
    pp. 677–726, 2021.
  ista: Bossmann L, Petrat SP, Pickl P, Soffer A. 2021. Beyond Bogoliubov dynamics.
    Pure and Applied Analysis. 3(4), 677–726.
  mla: Bossmann, Lea, et al. “Beyond Bogoliubov Dynamics.” <i>Pure and Applied Analysis</i>,
    vol. 3, no. 4, Mathematical Sciences Publishers, 2021, pp. 677–726, doi:<a href="https://doi.org/10.2140/paa.2021.3.677">10.2140/paa.2021.3.677</a>.
  short: L. Bossmann, S.P. Petrat, P. Pickl, A. Soffer, Pure and Applied Analysis
    3 (2021) 677–726.
corr_author: '1'
date_created: 2024-01-28T23:01:43Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2025-04-14T07:44:02Z
day: '01'
department:
- _id: RoSe
doi: 10.2140/paa.2021.3.677
ec_funded: 1
external_id:
  arxiv:
  - '1912.11004'
fulldoi: https://doi.org/10.2140/paa.2021.3.677
intvolume: '         3'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1912.11004
month: '10'
oa: 1
oa_version: Preprint
page: 677-726
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Beyond Bogoliubov dynamics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 3
year: '2021'
...
---
_id: '14891'
abstract:
- lang: eng
  text: We give the first mathematically rigorous justification of the local density
    approximation in density functional theory. We provide a quantitative estimate
    on the difference between the grand-canonical Levy–Lieb energy of a given density
    (the lowest possible energy of all quantum states having this density) and the
    integral over the uniform electron gas energy of this density. The error involves
    gradient terms and justifies the use of the local density approximation in the
    situation where the density is very flat on sufficiently large regions in space.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Mathieu
  full_name: Lewin, Mathieu
  last_name: Lewin
- first_name: Elliott H.
  full_name: Lieb, Elliott H.
  last_name: Lieb
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Lewin M, Lieb EH, Seiringer R. The local density approximation in density functional
    theory. <i>Pure and Applied Analysis</i>. 2020;2(1):35-73. doi:<a href="https://doi.org/10.2140/paa.2020.2.35">10.2140/paa.2020.2.35</a>
  apa: Lewin, M., Lieb, E. H., &#38; Seiringer, R. (2020). The local density approximation
    in density functional theory. <i>Pure and Applied Analysis</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/paa.2020.2.35">https://doi.org/10.2140/paa.2020.2.35</a>
  chicago: Lewin, Mathieu, Elliott H. Lieb, and Robert Seiringer. “The Local Density
    Approximation in Density Functional Theory.” <i>Pure and Applied Analysis</i>.
    Mathematical Sciences Publishers, 2020. <a href="https://doi.org/10.2140/paa.2020.2.35">https://doi.org/10.2140/paa.2020.2.35</a>.
  ieee: M. Lewin, E. H. Lieb, and R. Seiringer, “The local density approximation in
    density functional theory,” <i>Pure and Applied Analysis</i>, vol. 2, no. 1. Mathematical
    Sciences Publishers, pp. 35–73, 2020.
  ista: Lewin M, Lieb EH, Seiringer R. 2020. The local density approximation in density
    functional theory. Pure and Applied Analysis. 2(1), 35–73.
  mla: Lewin, Mathieu, et al. “The Local Density Approximation in Density Functional
    Theory.” <i>Pure and Applied Analysis</i>, vol. 2, no. 1, Mathematical Sciences
    Publishers, 2020, pp. 35–73, doi:<a href="https://doi.org/10.2140/paa.2020.2.35">10.2140/paa.2020.2.35</a>.
  short: M. Lewin, E.H. Lieb, R. Seiringer, Pure and Applied Analysis 2 (2020) 35–73.
corr_author: '1'
date_created: 2024-01-28T23:01:44Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2026-08-12T08:47:14Z
day: '01'
department:
- _id: RoSe
doi: 10.2140/paa.2020.2.35
external_id:
  arxiv:
  - '1903.04046'
fulldoi: https://doi.org/10.2140/paa.2020.2.35
intvolume: '         2'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1903.04046
month: '01'
oa: 1
oa_version: Preprint
page: 35-73
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: The local density approximation in density functional theory
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2
year: '2020'
...
---
_id: '6186'
abstract:
- lang: eng
  text: "We prove that the local eigenvalue statistics of real symmetric Wigner-type\r\nmatrices
    near the cusp points of the eigenvalue density are universal. Together\r\nwith
    the companion paper [arXiv:1809.03971], which proves the same result for\r\nthe
    complex Hermitian symmetry class, this completes the last remaining case of\r\nthe
    Wigner-Dyson-Mehta universality conjecture after bulk and edge\r\nuniversalities
    have been established in the last years. We extend the recent\r\nDyson Brownian
    motion analysis at the edge [arXiv:1712.03881] to the cusp\r\nregime using the
    optimal local law from [arXiv:1809.03971] and the accurate\r\nlocal shape analysis
    of the density from [arXiv:1506.05095, arXiv:1804.07752].\r\nWe also present a
    PDE-based method to improve the estimate on eigenvalue\r\nrigidity via the maximum
    principle of the heat flow related to the Dyson\r\nBrownian motion."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- 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: Dominik J
  full_name: Schröder, Dominik J
  id: 408ED176-F248-11E8-B48F-1D18A9856A87
  last_name: Schröder
  orcid: 0000-0002-2904-1856
citation:
  ama: 'Cipolloni G, Erdös L, Krüger TH, Schröder DJ. Cusp universality for random
    matrices, II: The real symmetric case. <i>Pure and Applied Analysis </i>. 2019;1(4):615–707.
    doi:<a href="https://doi.org/10.2140/paa.2019.1.615">10.2140/paa.2019.1.615</a>'
  apa: 'Cipolloni, G., Erdös, L., Krüger, T. H., &#38; Schröder, D. J. (2019). Cusp
    universality for random matrices, II: The real symmetric case. <i>Pure and Applied
    Analysis </i>. MSP. <a href="https://doi.org/10.2140/paa.2019.1.615">https://doi.org/10.2140/paa.2019.1.615</a>'
  chicago: 'Cipolloni, Giorgio, László Erdös, Torben H Krüger, and Dominik J Schröder.
    “Cusp Universality for Random Matrices, II: The Real Symmetric Case.” <i>Pure
    and Applied Analysis </i>. MSP, 2019. <a href="https://doi.org/10.2140/paa.2019.1.615">https://doi.org/10.2140/paa.2019.1.615</a>.'
  ieee: 'G. Cipolloni, L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality
    for random matrices, II: The real symmetric case,” <i>Pure and Applied Analysis
    </i>, vol. 1, no. 4. MSP, pp. 615–707, 2019.'
  ista: 'Cipolloni G, Erdös L, Krüger TH, Schröder DJ. 2019. Cusp universality for
    random matrices, II: The real symmetric case. Pure and Applied Analysis . 1(4),
    615–707.'
  mla: 'Cipolloni, Giorgio, et al. “Cusp Universality for Random Matrices, II: The
    Real Symmetric Case.” <i>Pure and Applied Analysis </i>, vol. 1, no. 4, MSP, 2019,
    pp. 615–707, doi:<a href="https://doi.org/10.2140/paa.2019.1.615">10.2140/paa.2019.1.615</a>.'
  short: G. Cipolloni, L. Erdös, T.H. Krüger, D.J. Schröder, Pure and Applied Analysis  1
    (2019) 615–707.
date_created: 2019-03-28T10:21:17Z
date_published: 2019-10-12T00:00:00Z
date_updated: 2026-07-29T13:46:19Z
day: '12'
department:
- _id: LaEr
doi: 10.2140/paa.2019.1.615
ec_funded: 1
external_id:
  arxiv:
  - '1811.04055'
fulldoi: https://doi.org/10.2140/paa.2019.1.615
intvolume: '         1'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1811.04055
month: '10'
oa: 1
oa_version: Preprint
page: 615–707
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
publication: 'Pure and Applied Analysis '
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: MSP
quality_controlled: '1'
related_material:
  record:
  - id: '6179'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: 'Cusp universality for random matrices, II: The real symmetric case'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 1
year: '2019'
...