Landauer-Büttiker formula and Schrödinger conjecture. (English) Zbl 1277.82055

The paper concerns transport in the so called electronic black box model. The model is used to describe a one-dimensional sample connected at its left and right ends to two infinitely extended reservoirs at distinct temperatures and chemical potentials. The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system (sample + reservoirs) in terms of scattering data. The resulting steady state entropy flux may vanish in the limit for the length of the sample going to infinity. The goal of the authors is to characterize the persistence of transport in this limit in terms of the spectral data of the limiting half-line Schrödinger operator of the system.


82C70 Transport processes in time-dependent statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
46N50 Applications of functional analysis in quantum physics
47N50 Applications of operator theory in the physical sciences
46L60 Applications of selfadjoint operator algebras to physics
46N55 Applications of functional analysis in statistical physics
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[1] Araki, H.: Relative entropy of states of von Neumann algebras. Publ. Res. Inst. Math. Sci. Kyoto Univ. 11, 809 (1975/76) · Zbl 0326.46031
[2] Avila, A.: On the Kotani-Last and Schrödinger conjectures. In preparation, available at http://arxiv.org/abs/1210.6325v1 [math.DS], 2012 · Zbl 0524.35002
[3] Aschbacher, W.; Jakšić, V.; Pautrat, Y.; Pillet, C.-A., Transport properties of quasi-free fermions, J. Math. Phys., 48, 032101, (2007) · Zbl 1137.82331
[4] Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Open Quantum System III. Recent Developments. S. Attal, A. Joye, C.-A. Pillet, editors. Lecture Notes in Mathematics 1882, New York: Springer, 2006, p. 1 · Zbl 1126.82032
[5] Gilbert, D.J.; Pearson, D., On subordinacy and analysis of the spectrum of one dimensional Schrödinger operators, J. Math. Anal., 128, 30, (1987) · Zbl 0666.34023
[6] Grech, P., Jakšić, V., Westrich M.: The spectral structure of the electronic black box hamiltonian. In preparation, available at http://arxiv.org/abs/1208.2420v1 [math-ph], 2012 · Zbl 1296.47005
[7] Jakšić, V.: Topics in spectral theory. In: Open Quantum Systems I. The Hamiltonian Approach. S. Attal, A. Joye, C.-A. Pillet, editors. Lecture Notes in Mathematics 1880, New York: Springer, 2006, p. 235 · Zbl 1051.82003
[8] Jakšić, V., Kritchevski, E., Pillet, C.-A.: Mathematical theory of the Wigner-Weisskopf atom. In: Large Coulomb systems. J. Dereziński, H. Siedentop, editors. Lecture Notes in Physics 695, New York: Springer, 2006, p. 145 · Zbl 1108.81054
[9] Jakšić, V.; Pillet, C.-A., On entropy production in quantum statistical mechanics, Commun. Math. Phys., 217, 285, (2001) · Zbl 1042.82031
[10] Kahn, S.; Pearson, D.B., Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta, 65, 505, (1992)
[11] Last, Y.; Simon, B., Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135, 329, (1999) · Zbl 0931.34066
[12] Maslov, V.P.; Molchanov, S.A.; Gordon, A.Ya, Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture, Russ. J. Math. Phys., 1, 71, (1993) · Zbl 0879.34078
[13] Nenciu, G., Independent electrons model for open quantum systems: Landauer-Büttiker formula and strict positivity of the entropy production, J. Math. Phys., 48, 033302, (2007) · Zbl 1137.82318
[14] Ruelle, D., Entropy production in quantum spin systems, Commun. Math. Phys., 224, 3, (2001) · Zbl 1051.82003
[15] Simon, B., Schrödinger semigroups, Bulletin AMS, 7, 447, (1982) · Zbl 0524.35002
[16] Simon, B., Bounded eigenfunctions and absolutely continuous spectra for one dimensional Schrödinger operators. proc, Amer. Math. Soc., 124, 3361, (1996) · Zbl 0869.34069
[17] Yafaev, D.R.: Mathematical scattering theory. General theory. Translated from the Russian by J. R. Schulenberger. Translations of Mathematical Monographs 105. Providence, RI: Amer. Math. Soc., 1992
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