Stability and absence of binding for multi-polaron systems

Rupert L. Frank; Elliott H. Lieb; Robert Seiringer; Lawrence E. Thomas

doi:10.1007/s10240-011-0031-5

Stability and absence of binding for multi-polaron systems

Rupert L. Frank, Elliott H. Lieb, Robert Seiringer, Lawrence E. Thomas

Mathematics

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18 Scopus citations

Abstract

We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich's 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling √α, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.

Original language	English (US)
Pages (from-to)	39-67
Number of pages	29
Journal	Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
Volume	113
Issue number	1
DOIs	https://doi.org/10.1007/s10240-011-0031-5
State	Published - Jun 2011

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s10240-011-0031-5

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title = "Stability and absence of binding for multi-polaron systems",

abstract = "We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fr{\"o}hlich's 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling √α, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.",

author = "Frank, {Rupert L.} and Lieb, {Elliott H.} and Robert Seiringer and Thomas, {Lawrence E.}",

note = "Funding Information: We are grateful to Herbert Spohn for making us aware of the problem of proving absence of binding for large U. We also thank Marcel Griesemer and Jacob Schach M{\o}ller for helpful comments on an early version of our manuscript. Partial financial support from the US National Science Foundation through grants PHY-0652854 (E. L. and R. F.) and PHY-0845292 (R. S.) are gratefully acknowledged. L. T. would like to thank the PIMS Institute, University of British Columbia, for their hospitality and support.",

year = "2011",

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doi = "10.1007/s10240-011-0031-5",

language = "English (US)",

volume = "113",

pages = "39--67",

journal = "Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques",

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T1 - Stability and absence of binding for multi-polaron systems

AU - Frank, Rupert L.

AU - Lieb, Elliott H.

AU - Seiringer, Robert

AU - Thomas, Lawrence E.

N1 - Funding Information: We are grateful to Herbert Spohn for making us aware of the problem of proving absence of binding for large U. We also thank Marcel Griesemer and Jacob Schach Møller for helpful comments on an early version of our manuscript. Partial financial support from the US National Science Foundation through grants PHY-0652854 (E. L. and R. F.) and PHY-0845292 (R. S.) are gratefully acknowledged. L. T. would like to thank the PIMS Institute, University of British Columbia, for their hospitality and support.

PY - 2011/6

Y1 - 2011/6

N2 - We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich's 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling √α, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.

AB - We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich's 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling √α, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.

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Stability and absence of binding for multi-polaron systems

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