# 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 critical temperature for the BCS equation at weak coupling. (English) Zbl 1137.82025
In this paper it is studied the Bardeen-Cooper-Schrieffer (BCS) equation for a Fermi gas at real chemical potential $\mu$ and positive temperature $T$, with local two-body interaction $\lambda V\left(x\right)$, where $\lambda$ denotes a coupling constant and $v\in {L}^{1}\left({ℝ}^{3}\right)\cap {L}^{3/2}\left({ℝ}^{3}\right)$. The main result of the present paper establishes the asymptotic behaviour of the critical temperature as $\lambda$ approaches 0. There are also obtained necessary and sufficient conditions on $V\left(x\right)$ for the existence of a nontrivial solution for all values of $\lambda >0$. The approach developed in this paper is not restricted to the kinetic energy ${K}_{T,\mu }$ appearing in the BCS model, but it can be adopted to any symbol vanishing on a manifold of codimension at least one.

##### MSC:
 82D55 Superconductors (statistical mechanics) 46N50 Applications of functional analysis in quantum physics 82D50 Superfluids (statistical mechanics) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
##### References:
 [1] Andrenacci, N., Perali, A., Pieri, P., and Strinati, G.C. Density-induced BCS to Bose-Einstein crossover,Phys. Rev. B 60, 12410, (1999). · doi:10.1103/PhysRevB.60.12410 [2] Bardeen, J., Cooper, L., and Schrieffer, J. Theory of superconductivity,Phys. Rev. 108, 1175–1204, (1957). · Zbl 0090.45401 · doi:10.1103/PhysRev.108.1175 [3] Bloch, I., Dalibard, J., and Zwerger, W. Many-body physics with ultracold gases, preprint, arXiv: (0704).3011. [4] Chen, Q., Stajic, J., Tan, S., and Levin, K. BCS-BEC crossover: From high temperature superconductors to ultracold superfluids.Phys. Rep. 412, 1–88, (2005). · doi:10.1016/j.physrep.2005.02.005 [5] Förster, C. Trapped modes for the elastic plate with a perturbation of Young’s modulus, preprint, arXiv:math-ph/0609032. [6] Förster, C. and Weidl, T. Trapped modes for an elastic strip with perturbation of the material properties,Quart. J. Mech. Appl. Math. 59, 399–418, (2006). · Zbl 1096.74023 · doi:10.1093/qjmam/hbl008 [7] Gor’kov, L. P., and Melik-Barkhudarov, T. K. Contributions to the theory of superfluidity in an imperfect Fermi gas,Soviet Physics JETP 13, 1018, (1961). [8] Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J.P. The BCS functional for general pair interactions,Commun. Math. Phys., in press. [9] Laptev, A., Safronov, O., and Weidl, T. Bound State Asymptotics for Elliptic Operators with Strongly Degenerated Symbols, inNonlinear Problems in Mathematical Physics and Related Topics I, 233–246, Int. Math. Ser. (N.Y.), Kluwer/Plenum, New York, (2002). [10] Leggett, A. J. Diatomic molecules and Cooper pairs, inModern Trends in the Theory of Condensed Matter, Pekalski, A. and Przystawa, R., Eds., Springer, (1980). [11] Lieb, E. and Loss, M.Analysis, American Mathematical Society, (2001). [12] Nozières, P. and Schmitt-Rink, S. Bose condensation in an attractive Fermion gas: From weak to strong coupling superconductivity.J. Low Temp. Phys. 59, 195–211, (1985). · doi:10.1007/BF00683774 [13] Parish, M., Mihaila, B., Timmermans, E., Blagoev, K., and Littlewood, P. BCS-BEC crossover with a finite-range interaction.Phys. Rev. B 71, 0645131–0645136, (2005). · doi:10.1103/PhysRevB.71.064513 [14] Randeria, M. Crossover from BCS Theory to Bose-Einstein Condensation, inBose-Einstein Condensation, Griffin, A., Snoke, D.W., and Stringari, S., Eds., Cambridge University Press, (1995). [15] Simon, B. The bound state of weakly coupled Schrödinger operators in one and two dimensions,Ann. Phys. 97, 279–288, (1976). · Zbl 0325.35029 · doi:10.1016/0003-4916(76)90038-5 [16] Sobolev, A. V. Asymptotic behavior of energy levels of a quantum particle in a homogeneous magnetic field perturbed by an attenuating electric field, I,Probl. Mat. Anal. 9, 67–84 (1984); II,Probl. Mat. Fiz. 11, 232–248, (1986).