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Supersymmetric generalized power functions. (English) Zbl 1454.81098

Summary: Complex-valued functions defined on a finite interval \([a, b]\) generalizing power functions of the type \((x - x_0)^n\) for \(n \geq 0\) are studied. These functions called \(\Phi\)-generalized powers, \( \Phi\) being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm-Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter [Math. Methods Appl. Sci. 33, No. 4, 459–468 (2010; Zbl 1202.34060)]. The \(\Phi\)-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking \({\Phi} = \psi_0^2\), where the function \(\psi_0(x)\) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as \(\Phi\)-symmetric conjugate and antisymmetry of the \(\Phi\)-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four \(\Phi\)-trigonometric \(( \Phi\)-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the \(\Phi\)-derivatives. We show that the first \(n \Phi\)-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order \(n + 1\). Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.
©2020 American Institute of Physics

MSC:

81Q60 Supersymmetry and quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B24 Sturm-Liouville theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Citations:

Zbl 1202.34060
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References:

[1] Kravchenko, V. V., A representation for solutions of the Sturm-Liouville equation, Complex Var. Elliptic Equations, 53, 8, 775-789 (2008) · Zbl 1183.30052
[2] Kravchenko, V. V.; Porter, R. M., Spectral parameter power series for Sturm-Liouville problems, Math. Methods Appl. Sci., 33, 4, 459-468 (2010) · Zbl 1202.34060
[3] Begehr, H. G. W.; Gilbert, R. P., Transformations, Transmutations and Kernel Functions (1992), Longman Scientific & Technical · Zbl 0827.35001
[4] Carroll, R. W., Transmutation Theory and Applications (1985), North-Holland · Zbl 0581.35004
[5] Colton, D. L., Solution of Boundary Value Problems by the Method of Integral Operators (1976), Pitman: Pitman, London · Zbl 0332.35001
[6] Levitan, B. M., Inverse Sturm-Liouville Problems (1987), VNU Science Press · Zbl 0749.34001
[7] Marchenko, V., Sturm-Liouville Operators and Applications (1986), Birkhäuser Basel
[8] Kravchenko, V. V., Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions, Appl. Math. Comput., 328, 75-81 (2018) · Zbl 1427.34023
[9] Kravchenko, V. V.; Morelos, S.; Torba, S. M., Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems, Appl. Math. Comput., 273, 321-336 (2016) · Zbl 1410.34071
[10] Kravchenko, V. V.; Torba, S. M., Transmutations and Spectral Parameter Power Series in Eigenvalue Problems, 209-238 (2013), Springer Basel: Springer Basel, Basel · Zbl 1347.34130
[11] Kravchenko, V. V.; Torba, S. M., Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems, J. Comput. Appl. Math., 275, 1-26 (2015) · Zbl 1337.65099
[12] Blancarte, H.; Campos, H. M.; Khmelnytskaya, K. V., Spectral parameter power series method for discontinuous coefficients, Math. Methods Appl. Sci., 38, 10, 2000-2011 (2015) · Zbl 1341.34035
[13] Erbe, L.; Mert, R.; Peterson, A., Spectral parameter power series for Sturm-Liouville equations on time scales, Appl. Math. Comput., 218, 14, 7671-7678 (2012) · Zbl 1245.34088
[14] Khmelnytskaya, K. V.; Kravchenko, V. V.; Rosu, H. C., Eigenvalue problems, spectral parameter power series, and modern applications, Math. Methods Appl. Sci., 38, 10, 1945-1969 (2014) · Zbl 1347.34128
[15] Khmelnytskaya, K. V.; Rosu, H. C., Spectral parameter power series representation for Hill’s discriminant, Ann. Phys., 325, 11, 2512-2521 (2010) · Zbl 1213.34027
[16] Porter, R. M., On Sturm-Liouville equations with several spectral parameters, Bole. Soc. Mat. Mex., 22, 1, 141-163 (2016) · Zbl 1344.34037
[17] Castillo-Pérez, R.; Kravchenko, V. V.; Torba, S. M., Spectral parameter power series for perturbed Bessel equations, Appl. Math. Comput., 220, 676-694 (2013) · Zbl 1329.34021
[18] Kravchenko, V. V.; Torba, S. M.; Castillo-Pérez, R., A neumann series of Bessel functions representation for solutions of perturbed Bessel equations, Appl. Anal., 97, 5, 677-704 (2018) · Zbl 1395.34016
[19] Rabinovich, V. S.; Hernández-Juárez, J., Method of the spectral parameter power series in problems of underwater acoustics of the stratified ocean, Math. Methods Appl. Sci., 38, 10, 1990-1999 (2015) · Zbl 1327.76128
[20] Khmelnytskaya, K. V.; Serroukh, I., The heat transfer problem for inhomogeneous materials in photoacoustic applications and spectral parameter power series, Math. Methods Appl. Sci., 36, 14, 1878-1891 (2013) · Zbl 1276.35098
[21] Rabinovich, V.; Urbano Altamirano, F., Application of the SPPS method to the one-dimensional quantum scattering, Commun. Math. Anal., 17, 295 (2007) · Zbl 1320.81088
[22] Castillo-Pérez, R.; Kravchenko, V. V.; Torba, S. M., Analysis of graded-index optical fibers by the spectral parameter power series method, J. Opt., 17, 2, 025607 (2015)
[23] Bilodeau, A.; Tremblay, S., On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators, J. Phys. A: Math. Theor., 46, 42, 425302 (2013) · Zbl 1276.81064
[24] Han, Z.; Hu, Y.; Lee, C., Optimal pricing barriers in a regulated market using reflected diffusion processes, Quant. Finance, 16, 4, 639-647 (2016)
[25] Khmelnytskaya, K. V.; Kravchenko, V. V.; Baldenebro-Obeso, J. A., Spectral parameter power series for fourth-order Sturm-Liouville problems, Appl. Math. Comput., 219, 8, 3610-3624 (2012) · Zbl 1311.34026
[26] Witten, E., Dynamical breaking of supersymmetry, Nucl. Phys. B, 188, 3, 513-554 (1981) · Zbl 1258.81046
[27] Chen, K.-T., Iterated path integrals, Bull. Am. Math. Soc., 83, 5, 831-879 (1977) · Zbl 0389.58001
[28] Brown, F., Iterated Integrals in Quantum Field Theory (2013), Cambridge University Press · Zbl 1295.81072
[29] Kravchenko, V. V., On the completeness of systems of recursive integrals, Commun. Math. Anal., 2011, 3, 172-176 · Zbl 1221.46013
[30] Kravchenko, V. V.; Morelos, S.; Tremblay, S., Complete systems of recursive integrals and taylor series for solutions of Sturm-Liouville equations, Math. Methods Appl. Sci., 35, 6, 704-715 (2011) · Zbl 1243.34011
[31] Davis, P. J., Interpolation and Approximation (1975), Dover Publications
[32] Goursat, E., Cours d’Analyse Mathématique, Vol. 2: Théorie des Fonctions Analytiques; Équations Différentielles; Équations aux Dérivées Partielles du Premier Ordre (Classic Reprint) (2018), Fb&c Limited
[33] Widder, D. V., A generalization of Taylor’s series, Trans. Am. Math. Soc., 30, 1, 126-154 (1928) · JFM 54.0251.05
[34] Widder, D. V., On the expansion of analytic functions of the complex variable in generalized Taylor’s series, Trans. Am. Math. Soc., 31, 1, 43-52 (1929) · JFM 55.0180.03
[35] Volterra, V., Theory of Functionals and of Integral and Integro-Differential Equations (2005), Courier Corporation
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