New exact representations for the solutions of numerous one-dimensional spectral problems for the Schrödinger operator with polynomial potential are obtained by using a technique based on the functional-discrete (FD) method. In the cases where the ordinary FD-method is divergent, we propose to use its modification, which proves to be quite efficient. The obtained theoretical results are illustrated by numerical examples.
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References
E. Magyari, “Exact quantum-mechanical solutions for anharmonic oscillators,” Phys. Lett. A, 81, No. 2–3, 116–118 (1981).
K. Banerjee, “General anharmonic oscillators,” Proc. Roy. Soc. London A, 364, No. 1717, 263–275 (1978).
R. N. Chaudhuri and M. Mondal, “Improved Hill determinant method: general approach to the quantum anharmonic oscillators,” Phys. Rev. A, 43, No. 7, 3241–3246 (1991).
R. Adhikari, R. Dutt, and Y. P. Varshni, “Exact solutions for polynomial potentials using supersymmetry inspired factorization method,” Phys. Lett. A, 141, No. 1–2, 1–8 (1981).
Y.-M. Kao and T. F. Jiang, “Adomian’s decomposition method for eigenvalue problems,” Phys. Rev. E, 71, 036702 (2005).
A. K. Roy, N. Gupta, and B. M. Deb, “Time-dependent quantum-mechanical calculation of ground and exited states of anharmonic and double-well oscillators,” Phys. Rev. A, 65, No. 1, 012109 (2001).
A. Ronveaux (editor), Heun’s Differential Equations, Oxford Univ. Press, Oxford (1995).
V. L. Makarov, “FD-method in spectral problems for the Schrödinger operator with polynomial potential on (−∞, +∞),” Dop. Nats. Akad. Nauk Ukr., No. 11, 5–11 (2015).
V. L. Makarov, “On a functional-difference method of any order of accuracy for the solution of the Sturm–Liouville problem with piecewise smooth coefficients,” Dokl. Akad. Nauk SSSR, 1 (320), 34–39 (1991).
V. L. Makarov, “FD-method—exponential rate of convergence,” Zh. Obchysl.. Prykl. Mat., No. 82, 69–74 (1997).
G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Dordrecht (1994).
V. L. Makarov and N. M. Romanyuk, “New properties of the FD-method in its application to Sturm–Liouville problems,” Dop. Nats. Akad. Nauk Ukr., No. 2, 26–31 (2014).
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (editors), NIST Digital Library of Mathematical Functions, Cambridge Univ. Press, New York (2010).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1954).
A. Zettl, Sturm–Liouville Theory, American Mathematical Society, Providence, RI (2005).
J. D. Pryce, Numerical Solution of Sturm–Liouville Problems, Oxford Univ. Press, Oxford (1993).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 1, pp. 79–93, January, 2018.
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Makarov, V.L. Exact and Approximate Solutions of Spectral Problems for the Schrödinger Operator on (−∞, ∞) with Polynomial Potential. Ukr Math J 70, 84–100 (2018). https://doi.org/10.1007/s11253-018-1489-9
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DOI: https://doi.org/10.1007/s11253-018-1489-9