The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation

J. Bryce McLeod, Yisong Yang

Research output: Contribution to journalArticle

Abstract

In this paper we study the Bardeen-Cooper-Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.

Original languageEnglish (US)
Pages (from-to)6007-6025
Number of pages19
JournalJournal of Mathematical Physics
Volume41
Issue number9
StatePublished - Sep 2000

Fingerprint

uniqueness
Positive Solution
Uniqueness
Bounded Domain
Approximation
approximation
Energy Gap
Nonuniqueness
Superconductor
Phonon
Finite Temperature
Kernel Function
Critical Temperature
Existence and Uniqueness
Phase Transition
kernel
Sufficient Conditions
kernel functions
Zero
critical temperature

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation. / McLeod, J. Bryce; Yang, Yisong.

In: Journal of Mathematical Physics, Vol. 41, No. 9, 09.2000, p. 6007-6025.

Research output: Contribution to journalArticle

@article{62939e5c4d0b43c5b62bd8883749277e,
title = "The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation",
abstract = "In this paper we study the Bardeen-Cooper-Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.",
author = "McLeod, {J. Bryce} and Yisong Yang",
year = "2000",
month = "9",
language = "English (US)",
volume = "41",
pages = "6007--6025",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics Publising LLC",
number = "9",

}

TY - JOUR

T1 - The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation

AU - McLeod, J. Bryce

AU - Yang, Yisong

PY - 2000/9

Y1 - 2000/9

N2 - In this paper we study the Bardeen-Cooper-Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.

AB - In this paper we study the Bardeen-Cooper-Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.

UR - http://www.scopus.com/inward/record.url?scp=0034345862&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034345862&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034345862

VL - 41

SP - 6007

EP - 6025

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

ER -