Abstract
The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.
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Haken, H.: Synergetics. Berlin, Heidelberg, New York: Springer 1977
Agarwal, G.S.: Phys. Rev. A1, 1445 (1970)
Voigt, H., Bandilla, A., Ritze, H.H.: Z. Physik B36, 295 (1980)
Sargent III, M., Scully, M.O., Lamb, W.E.: Laser Physics. p. 285. London, Amsterdam, Don Mills, Sidney, Tokyo: Addison Wesley Publishing company 1974
McQuarrie, D.A.: J. Appl. Probab.4, 413 (1967)
Görtz, R., Walls, D.F.: Z. Physik B25, 423 (1976)
Nicolis, G.: J. Statist. Phys.6, 195 (1972)
Dixit, S.N., Zoller, P., Lambropoulos, P.: Phys. Rev. A21, 1289 (1980)
Dekker, H., van Kampen, N.G.: Phys. Lett. A73, 374 (1979)
Graffi, S., Grecchi, V.: Lett. Nuovo Cimento12, 425 (1975)
Denk, H.: to be published
Risken, H., Vollmer, H.D.: Z. Physik B39, 89 (1980)
Tikhonov, V.I.: Avtom. Telemekh.21, 301 (1960)
Risken, H., Vollmer, H.D.: Z. Physik B31, 209 (1978)
Risken, H., vollmer, H.D.: Z. Physik B33, 297 (1979)
Vollmer, H.D., Risken, H.: Z. Physik B34, 313 (1979)
Brinkman, H.C.: Physica22, 29 (1956)
Risken, H., Vollmer, H.D., Denk, H.: Phys. Lett. A (to be published)
Risken, H., Vollmer, H.D.: Z. Physik201, 323 (1967)
Hempstead, R.D., Lax, M.: Phys. Rev.161, 350 (1967)
Risken, H.: Fortschr. Phys.16, 261 (1968)
Haken, H.: Handbuch der Physik. Bd. XXV/2c. Berlin, Heidelberg, New York: Springer 1970
Risken, H.: In: Progress in Optics. vol. VIII, Wolf, E. (ed.), vol. VIII, p. 239. Amsterdam: North-Holland 1970
Seybold, K., Risken, H.: Z. Physik267, 323 (1974)
Grossmann, S.: Phys. Rev. A17, 1123 (1978)
King, H., Deker, U., Haake, F.: Z. Physik B36, 205 (1979)
Ziegler, K., Horner, H.: Z. Physik B37, 339 (1980)
Haag, G., Hänggi, P.: Z. Physik B34, 411 (1979)
Denk, H., Riederle, M.: (to be published)
Kenkre, V.M.: In: Statistical Mechanics and Statistical Methods in Theory and Application. Landmann, U. (ed.), p. 441. New York, London: Plenum Press 1977
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover Publications 1965
Perron, O.: Die Lehre von den Kettenbrüchen. Bd. II, p. 47. Stuttgart: B.G. Teubner 1957
Wall, H.S.: Analytic Theory of Continued Fractions. p. 353. Bronx, N.Y.: Chelsea Publishing Company 1967
Gradsteyn, I.S., Ryzhik, M.: Table of Integrals Series and Products, p. 315. New York: Academic Press 1965
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Risken, H., Vollmer, H.D. Solutions and applications of tridiagonal vector recurrence relations. Z. Physik B - Condensed Matter 39, 339–346 (1980). https://doi.org/10.1007/BF01305834
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DOI: https://doi.org/10.1007/BF01305834