FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Application of the Lagrange formula to calculate the eigenvalues of a harmonic chain with dissipation at the boundaries |
Gudimenko A.I., Likhosherstov A.V. |
2025, issue 2, P. 232–243 DOI: https://doi.org/10.47910/FEMJ202516 |
Abstract |
| The eigenvalue problem for a dynamic system describing in Schrodinger coordinates the oscillations of a homogeneous harmonic chain with dissipation at the boundaries is considered. The combinatorial Lagrange formula is used to obtain a uniform approximation of the eigenvalues for a sufficiently large number of particles in the chain. |
Keywords: harmonic chain, Schrodinger coordinates, eigenvalue problem. |
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References |
| [1] Schrodinger E., “Zur Dynamik elastisch gekoppelter Punktsysteme”, Annalen der Physik, 44, (1914), 916–934. [2] Takizawa E., Kobayasi K., “Heat Flow in a System of Coupled Harmonic Oscillators”, Chinese J. Phys., 1:2, (1963), 59–73. [3] Gudimenko A.I., Likhosherstov A., “Spectral problem for a harmonic chain with dissipation at the boundaries”, Math. Notes, 116:4, (2024), 600–613. [4] Comtet L., Advanced combinatorics. The art of finite and infinite expansions, D. Reidel Publishing Company, Dordrecht-Holland, 1974. [5] Goursat E., A Course in Mathematical Analysis, Vol. 2: Functions of a Complex Variable, Ginn and Compaty, Boston, New York, 1916. [6] Whittaker E. T., Watson G. N., A Course of Modern Analysis, Cambridge University Press, Cambridge, UK, 2021. [7] Losonczi L., “Eigenvalues and eigenvectors of some tridiagonal matrices”, Acta Math. Hun-gar., 60, (1992), 309–322. [8] da Fonseca C.M., Kowalenko V., “Eigenpairs of a family of tridiagonal matrices: three decades later”, Acta Math. Hungar., 160, (2020), 376–389. [9] Du Z., da Fonseca C. M., “Root location for the characteristic polynomial of a Fibonacci type sequence”, Czechoslovak Mathematical Journal, 73:1, (2023), 189–195. [10] Losonczi L., “On the zeros of reciprocal polynomials”, Publ. Math. Debrecen., 94:3–4, (2019), 455–466. [11] Rieder Z., Lebowitz J. L., Lieb E., “Properties of a harmonic crystal in a stationary nonequilibrium state”, J. Math. Phys., 8:5, (1967), 1073–1078. [12] Nakazawa H., “On the lattice thermal conduction”, Progress of Theoretical Physics Sup-plement, 45, (1970), 231–262. [13] Weiderpass G.A., Monteiro G.M., Caldeira A.O., “Exact solution for the heat conductance in harmonic chains”, Phys. Rev. B, 102, (2020), 125401. [14] Guzev M.A., Dmitriev A.A., “Razlichnye formy predstavleniya resheniya odnomernoi garmonicheskoi modeli kristalla”, Dal'nevost. matem. zhurn., 17:1, (2017), 30–47. [15] Evgrafov M.A., Analiticheskie funktsii, Nauka GRFML, Moskva, 1991. |