zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The solution to the BCS gap equation and the second-order phase transition in superconductivity. (English) Zbl 1223.82074
The first part of the present paper is devoted to an alternative proof of the existence of a unique solution to the BCS gap equation. Next, it is defined a certain subspace W of a Banach space consisting of continuous functions and it is considered the solution approximated by an element of W. Next, it is shown that, under this approximation, the transition to a superconducting state is a second-order phase transition. In other words, it is established that the condition that the solution belongs to W is a sufficient condition for the second-order phase transition to superconductivity.
MSC:
82D55Superconductors (statistical mechanics)
82B26Phase transitions (general)
45G10Nonsingular nonlinear integral equations
References:
[1]Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044 · doi:10.1137/1018114
[2]Bardeen, J.; Cooper, L. N.; Schrieffer, J. R.: Theory of superconductivity, Phys. rev. 108, 1175-1204 (1957) · Zbl 0090.45401
[3]Billard, P.; Fano, G.: An existence proof for the gap equation in the superconductivity theory, Comm. math. Phys. 10, 274-279 (1968) · Zbl 0164.57002
[4]Bogoliubov, N. N.: A new method in the theory of superconductivity I, Soviet phys. JETP 34, 41-46 (1958)
[5]Frank, R. L.; Hainzl, C.; Naboko, S.; Seiringer, R.: The critical temperature for the BCS equation at weak coupling, J. geom. Anal. 17, 559-568 (2007) · Zbl 1137.82025 · doi:10.1007/BF02937429
[6]Hainzl, C.; Hamza, E.; Seiringer, R.; Solovej, J. P.: The BCS functional for general pair interactions, Comm. math. Phys. 281, 349-367 (2008) · Zbl 1161.82027 · doi:10.1007/s00220-008-0489-2
[7]Hainzl, C.; Seiringer, R.: Critical temperature and energy gap for the BCS equation, Phys. rev. B 77, 184517 (2008)
[8]Niwa, M.: Fundamentals of superconductivity, (2002)
[9]Odeh, F.: An existence theorem for the BCS integral equation, IBM J. Res. develop. 8, 187-188 (1964)
[10]Vansevenant, A.: The gap equation in the superconductivity theory, Phys. D 17, 339-344 (1985)
[11]Watanabe, S.: Superconductivity and the BCS-Bogoliubov theory, JP J. Algebra number theory appl. 11, 137-158 (2008) · Zbl 1225.82088 · doi:http://www.pphmj.com/cart.php?act=cart_add&cart_link_type=article&cart_art_id=3362
[12]Watanabe, S.: A mathematical proof that the transition to a superconducting state is a second-order phase transition, Far east J. Math. sci. 34, 37-57 (2009) · Zbl 1178.82093 · doi:http://pphmj.com/abstract/4155.htm
[13]Watanabe, S.: Is the solution to the BCS gap equation continuous in the temperature?
[14]Zeidler, E.: Applied functional analysis, Appl. math. Sci. 108 (1995)
[15]Ziman, J. M.: Principles of the theory of solids, (1972)