Cyclic Operator Decomposition for Solving the Differential Equations

Affiliation(s)

Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia.

Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia.

ABSTRACT

We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the generating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the perturbation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classical oscillator, Schrodinger equation, and wave equation in dispersive medium.

Cite this paper

I. Gonoskov, "Cyclic Operator Decomposition for Solving the Differential Equations,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 178-182. doi: 10.4236/apm.2013.31A025.

I. Gonoskov, "Cyclic Operator Decomposition for Solving the Differential Equations,"

References

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[13] B. Curgus and T. T. Read, “Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems,” Journal of Differential Equations, Vol. 184, No. 2, 2002, pp. 526-548. doi:10.1006/jdeq.2001.4152

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[17] L. Landau and E. Lifshitz, “Electrodynamics of Continuous Media,” Vol. 8, Butterworth-Heinemann, Oxford, 1984.

[1] F. J. Dyson, “Divergence of Perturbation Theory in Quantum Electrodynamics,” Physical Review, Vol. 85, No. 4, 1952, pp. 631-632. doi:10.1103/PhysRev.85.631

[2] A. V. Turbiner, “The Eigenvalue Spectrum in Quantum Mechanics and the Nonlinearization Procedure,” Soviet Physics Uspekhi, Vol. 27, No. 9, 1984, p. 668. doi:10.1070/PU1984v027n09ABEH004155

[3] J. Fleck, J. Morris and M. Feit, “Time-Dependent Propagation of High Energy Laser Beams through the Atmosphere,” Applied Physics A: Materials Science and Processing, Vol. 10, No. 2, 1976, pp. 129-160. doi:10.1007/BF00896333

[4] A. N. Drozdov and S. Hayashi, “Numerical Test of Approximate Single-Step Propagators: Harmonic Power Series Expansions versus System-Specific Split Operator Representations,” Physical Review E, Vol. 59, No. 2, 1999, pp. 1386-1397. doi:10.1103/PhysRevE.59.1386

[5] J. V. Corbett and J. Math, “Convergence of the Born Se- ries,” Journal of Mathematical Physics, Vol. 9, No. 6, 1968, p. 891. doi:10.1063/1.1664655

[6] M. Wellner, “Improvement of the Born Series at Low Energy,” Physical Review, Vol. 132, No. 4, 1963, pp. 1848-1853. doi:10.1103/PhysRev.132.1848

[7] R. Perez and J. Math, “On the Expansion of the Propagator in Power Series of the Coupling Constant,” Journal of Mathematical Physics, Vol. 20, No. 2, 1979, p. 241. doi:10.1063/1.524070

[8] F. S. Bemfica and H. O. Girotti, “Born Series and Unitarity in Noncommutative Quantum Mechanicsphys,” Physical Review D, Vol. 77, No. 2, 2008, Article ID: 027704. doi:10.1103/PhysRevD.77.027704

[9] F. J. Dyson, “The S Matrix in Quantum Electrodynamics,” Physical Review, Vol. 75, No. 11, 1949, pp. 1736- 1755. doi:10.1103/PhysRev.75.1736

[10] J. H. Eberly and H. R. Reiss, “Electron Self-Energy in Intense Plane-Wave Field,” Physical Review, Vol. 145, No. 4, 1966, pp. 1035-1040. doi:10.1103/PhysRev.145.1035

[11] H. R. Reiss and J. H. Eberly, “Green’s Function in Intense-Field Electrodynamics,” Physical Review, Vol. 151, No. 4, 1966, pp. 1058-1066. doi:10.1103/PhysRev.151.1058

[12] A. Mockel and J. Math, “Invariant Imbedding as a Generalization of the Resolvent Equation,” Journal of Mathematical Physics, Vol. 8, No. 12, 1967, p. 2318. doi:10.1063/1.1705158

[13] B. Curgus and T. T. Read, “Discreteness of the Spectrum of Second-Order Differential Operators and Associated Embedding Theorems,” Journal of Differential Equations, Vol. 184, No. 2, 2002, pp. 526-548. doi:10.1006/jdeq.2001.4152

[14] L. Landau and E. Lifshitz, “Quantum Mechanics Non-Relativistic Theory,” 3rd Edition, Pergamon Press, Oxford, 1977.

[15] L. Schiff, “Quantum Mechanics,” 3rd Edition, McGraw Hill, New York, 1968.

[16] F. J. Dyson, “The Schrodinger Equation in Quantum Electrodynamics,” Physical Review, Vol. 83, No. 6, 1951, pp. 1207-1216.

[17] L. Landau and E. Lifshitz, “Electrodynamics of Continuous Media,” Vol. 8, Butterworth-Heinemann, Oxford, 1984.