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  • 1
    Online Resource
    Online Resource
    Berlin [u.a.] : Springer
    Part of " Lecture notes in mathematics"
    Keywords: Hamilton-Operator ; Quasiklassisches Modell ; Hamilton-Operator ; Hamilton-Operator ; Halbklassische Behandlung ; Hamilton-Operator ; Quasiklassisches Modell ; Hamilton-Operator ; Hamilton-Operator ; Halbklassische Behandlung
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource
    ISBN: 9783540500766 , 9783540459132
    Series Statement: Lecture notes in mathematics 1336
    DDC: 515
    Language: English
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 187 (1997), S. 81-113 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: This paper is devoted to problems coming from statistical mechanics. The transfer matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator. Sometimes the new problem appears to have a semi-classical nature. Although the problem is similar to the semiclassical study of Kac's operator presented in our paper with M. Brunaud [BruHe] which was devoted to the study of for h small, new features appear for the model . Our first results concern semi-classical analysis of the ground state for this second operator. We then analyze the two models in the large dimension situation. One basic technique is Sjöstrand's formalism of the 0-standard functions as introduced in [Sj5]. The one-dimensional case was presented in [He11].
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 173 (1995), S. 335-356 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In the last years several contributions have been done around the total bandwidth of the spectrum for the Harper's operator. In particular an interesting conjecture has been proposed by Thouless which gives also strong convincing arguments for the proof in special cases. On the other hand, in the study of the Cantor structure of the spectrum, B. Helffer and J. Sjöstrand have justified an heuristic semiclassical approach proposed by M. Wilkinson. The aim of this article is to explain how one can use the first step of this approach to give a rigorous semi-classical proof of the Thouless formula in some of the simplest cases. We shall also indicate how one can hope with more effort to prove rigorously recent results of Last and Wilkinson on the same conjecture.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 201 (1999), S. 445-460 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We study the energy of relativistic electrons and positrons interacting through the second quantized Coulomb interaction and a self-generated magnetic field. As states we allow generalized Hartree–Fock states in the Fock space. Our main result is the assertion of positivity of the energy, if the atomic numbers and the fine structure constant are not too big. We also discuss the dependence of the result on the dressing of the electrons (choice of subspaces defining the electrons).
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 161 (1994), S. 631-643 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ2/μ1 between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L2(ℝ m ) by exp −V(x)/2 · exp Δx · exp −V(x)/2 where Δx is the Laplacian on ℝ m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ〉0.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Inventiones mathematicae 81 (1985), S. 205-216 
    ISSN: 1432-1297
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Letters in mathematical physics 38 (1996), S. 297-312 
    ISSN: 1573-0530
    Keywords: 81Q20 ; 82B21 ; Laplace integrals ; semi-classical analysis ; WKB constructions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This Letter gives detailed proofs concerning the analysis of the pair correlations for a nonconvex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operator. Although the problem is similar to the semiclassical study of the Kac operator presented in a paper with M. Brunaud, which was devoted⋆⋆ to the study of exp-(v/2) exp h 2 Δ exp-(v/2) for h small, new features appear for the model exp-(v/2h) exp hΔ exp-(v/2h). Our principal results concern the splitting of this operator between the two largest eigenvalues. We give an upper and a lower bound for this splitting in the semi-classical regime. As a corollary, we get good control of the decay of the pair correlation. Some of the results were announced in a previous paper. Related WKB constructions will be developed in a later paper.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Journal of statistical physics 74 (1994), S. 349-409 
    ISSN: 1572-9613
    Keywords: Statistical mechanics correlation ; thermonamic limit magnetization ; maximum principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The aim of this paper is to stu the behavior asm tends to ∞ of a family of measures exp[-Φ (m)(x)]dx (m) on ℝ m , whereΦ (m) is a potential on ℝ m which is a perturbation “in a suitable sense” of the harmonic potential Σ j x j 2 .
    Type of Medium: Electronic Resource
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