Abstract
We study the Landauer resistivity of the Kronig–Penney model which has various behavior depending on the potential and the Fermi energy. In the case of the Sturmian quasiperiodic potential, we discuss examples in which lim inf of it is zero.
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REFERENCES
F. Gesztesy and W. Kirsch, One-dimensional Schrödinger operators with interactions singular on a discrete set, J. Reine Angrew. Math. 362:28-50 (1985).
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Springer-Verlag, 1988).
M. Buttiker, Y. Imry, R. Landauer, and S. Pinhas, Generalized many-channel conductance formula with application to small rings, Phys. Rev. B 31:6207-6215 (1985).
J. Bellissard, A. Formoso, R. Lima, and D. Testard, Quasiperiodic interaction with a metal-insulator transition, Phys. Rev. B 26:3024-3030 (1982).
M. Kaminaga and F. Nakano, Spectral properties of quasiperiodic Kronig-Penney model, Tukuba J. of Math. 26:205-228 (2002).
W. Kirsch and F. Martinelli, On the spectrum of Schrödinger operators with a random potential, Commun. Math. Phys. 85:329-350 (1982).
P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators (Birkhäuser, 1985).
H. Schultz-Baldes and J. Bellissard, Anomalous transport: A mathematical framework, Rev. Math. Phys. 10:1-46 (1998).
H. Schultz-Baldes and J. Bellissard, A kinetic theory for quantum transport in aperiodic media, J. Stat. Phys. 91:991-1026 (1998).
F. Nakano and M. Kaminaga, Absence of transport under a slowly varying potential in disordered systems, J. Stat. Phys. 97:917-940 (1999).
F. Nakano, Absence of transport in Anderson localization, Rev. Math. Phys. 14:375-407 (2002).
A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Commun. Math. Phys. 111:409-415 (1987).
A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys. 56:525-531 (1989).
J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one dimensional quasi-crystals, Commun. Math. Phys. 125:527-543 (1989).
D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207:687-696 (1999).
M. Kaminaga and F. Nakano, Spectral properties of quasiperiodic Kronig-Penney model II.
S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183:171-189 (1999).
S. Jitomirskaya and Y. Last, Power law subordinacy and singular spectra. II. Line operators, Comm. Math. Phys. 211:643-658 (2000).
D. Damanik, R. Killip, and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals. III. α-continuity, Comm. Math. Phys. 212:191-204 (2000).
S. Lang, Introduction to Diophantine Approximations (Addison-Wesley, 1966).
B. Sutherland and M. Kohmoto, Resistance of a one-dimensional quasicrystal: Power-law growth, Phys. Rev. B 36:5877-5886 (1987).
B. Iochum and D. Testard, Power low growth for the resistance in the Fibonacci model, J. Stat. Phys. 65:715-723 (1991).
B. Iochum, L. Raymond, and D. Testard, Resistance of one-dimensional quasicrystals, Physica A 187:353-368 (1992).
D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals. II. The Lyapunov exponent, Lett. Math. Phys. 50:245-257 (1999).
M. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore en dimension 2, Commun. Math. Helv. 58:453-502 (1983).
H. L. Cycon, R. G. Froese, W. Kirsh, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics (Springer-Verlag, 1987).
J. E. Avron and B. Simon, Almost periodic Schrödinger operators II. The integrated density of states, Duke Math. J. 50:369-391 (1983).
J. M. Combes and L. E. Thomas, Asymptotic behavior of eigenfunctions for multi-particle Schrödinger operators, Commun. Math. Phys. 34:251-270 (1973).
D. Damanik, α‐continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192:169-182 (1998).
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Kaminaga, M., Nakano, F. The Landauer Resistivity on Quantum Wires. Journal of Statistical Physics 111, 339–353 (2003). https://doi.org/10.1023/A:1022265226479
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DOI: https://doi.org/10.1023/A:1022265226479