Far Eastern Mathematical Journal

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Harmonic oscillator chains with exactly solvable dynamics


Gudimenko A.I.

2017, issue 1, P. 11-21


Abstract
The method of Darboux transformation is applied to construct exactly solvable one-dimensional chains of harmonic oscillators.

Keywords:
Darboux transformation, exactly solvable dynamics, harmonic chains

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References

[1] A.A. Maradudin, E.W Montroll, G.H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press, New York, 1963.
[2] A. Dhar, K. Saito, “Heat Transport in Harmonic Systems”, Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer, Lecture Notes in Physics, v. 921, eds. S. Lepri, Springer, 2016, 39–105.
[3] I. Fujiwara, P.C. Hemmer, H. Wergeland, “Some Exact Results in the Theory of Brownian Motion”, Prog. Theor. Phya. Suppl., 37–38, (1966), 141–152.
[4] R.J. Rubin, “Momentum Autocorrelation Function of a Heavy Particle in a Finite Crystal”, Journal of the American Chemical Society, 90:12, (1968), 3061–3063.
[5] M.B.Yu, “Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses III. Some limiting cases”, Physica A, 447, (2016), 411–421.
[6] P. Mazur, E. Montroll, “Poincare Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators”, J. Math. Phys., 1:1, (1960), 70–84.
[7] M.H. Lee, “Local Dynamics in an Infinite Harmonic Chain”, Symmetry, 8:22, (2016), 1–12.
[8] M.A. Huerta, H.S. Robertson, “Entropy, Information Theory, and the Approach to Equilibrium of Coupled Harmonic Oscillator Systems”, J. Stat. Phys., 1:3, (1969), 393–414.
[9] A.J. O’Connor, “A Central Limit Theorem for the Disordered Harmonic Chain”, Commun. math. Phys., 45, (1975), 63–77.
[10] Z. Rieder, J.L. Lebowitz, E. Lieb, “Properties of a harmonic crystal in a stationary nonequilibrium state”, J. Math. Phys., 8:5, (1967), 1073–1078.
[11] R.J. Rubin, W.L. Greer, “Abnormal Lattice Thermal Conductivity of a One-Dimensional, Harmonic, Isotopically Disordered Crystal”, J. Math. Phys., 12:8, (1971), 1686–1701.
[12] F. Bonetto, J.L. Lebowitz, L. Rey-Bellet, “Fourier law: a challenge to theorists”, Mathematical Physics 2000, eds. A. Fokas, et al., Imperial College Press, London, 2000, 128.
[13] S. Lepri, R. Livi, A. Politi, “Heat Transport in Low Dimensions: Introduction and Phenomenology”, Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer, Lecture Notes in Physics, v. 921, eds. S. Lepri, Springer, 2016, 1–37.
[14] C.-W. Chang, “Experimental Probing of Non-Fourier Thermal Conductors”, Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer, Lecture Notes in Physics, v. 921, eds. S. Lepri, Springer, 2016, 305–338.
[15] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991.
[16] V. Spiridonov, A. Zhedanov, “Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials”, Methods and Applications of Analysis, 2:4, (1995), 369–398.
[17] V. Spiridonov, A. Zhedanov, “Discrete Reflectionless Potentials, Quantum Algebras, and q-Orthogonal Polynomials”, Annals of physics, 237, (1995), 126–146.

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