A spectral theory for the fractional memory Helmholtz operator and its application to quantum systems
DOI:
https://doi.org/10.62059/6y2dxw63Keywords:
Fractional memory Helmholtz operator, Spectral theory, Fractional calculus, Helmholtz equation, AFA operator, Eigenvalue problems, Quantum mechanics, Temporal memoryAbstract
We introduce the Fractional Memory Helmholtz Operator (FMHO), a generalized spectral operator constructed from Romero’s AFA operator for the description of wave phenomena with temporal memory. The proposed formulation extends the classical Helmholtz framework by embedding fractional temporal dynamics directly into the spectral operator rather than modifying the governing equations. A generalized Helmholtz eigenvalue problem is formulated and shown to admit, under separable solutions, a decomposition of its spectrum into a classical Helmholtz contribution and a memory-induced fractional correction. This result provides a natural interpretation of temporal memory as an intrinsic spectral property of the operator. As a direct consequence, a generalized Schr¨odinger–Helmholtz equation is obtained and applied to the one-dimensional infinite quantum well. The analysis demonstrates that the spatial eigenfunctions coincide with the classical solutions, while the temporal evolution is governed by the Mittag–Leffler function, producing a shift of the energy spectrum associated with temporal memory. The proposed framework establishes a unified operator-based formulation for generalized Helmholtz problems and opens new perspectives for the mathematical modeling of non-Markovian phenomena in quantum mechanics, wave propagation, optics, acoustics, and other complex systems.
References
[1] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2018).
[2] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2021).
[3] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed. (Academic Press, Amsterdam, 2013).
[4] L. C. Evans, Partial Differential Equations, 2nd ed. (American Mathematical Society, Providence, RI, 2010).
[5] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. I: Functional Analysis (Academic Press, New York, 1980).
[6] T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1995).
[7] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York, 1953).
[8] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).
[9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).
[10] K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Berlin, 2010).
[11] K. B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).
[12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010).
[13] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach, Amsterdam, 1993).
[14] M. Caputo, Linear models of dissipation whose Q is almost frequency independent–II, Geophys. J. Int. 13, 529–539 (1967).
[15] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion, Phys. Rep. 339, 1–77 (2000).
[16] R. Metzler and J. Klafter, The restaurant at the end of the random walk, J. Phys. A 37, R161–R208 (2004).
[17] N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62, 3135–3145 (2000).
[18] N. Laskin, Fractional Schr¨odinger equation, Phys. Rev. E 66, 056108 (2002).
[19] G. M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C. R. Acad. Sci. Paris 137, 554–558 (1903).
[20] H. von Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, ¨ J. Reine Angew. Math. 55, 25–55 (1860).
[21] L. G. Romero, A generalization of the Laplacian differential operator, Palestine Journal of Mathematics 5 (Special Issue), 24–33 (2016).
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Copyright (c) 2026 Luis Guillermo Romero (Autor/a)
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