Saigo-Maeda Operators Involving the Appell Function, Real Spectra from Symmetric Quantum Hamiltonians and Violation of t
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Saigo-Maeda Operators Involving the Appell Function, Real Spectra from Symmetric Quantum Hamiltonians and Violation of the Second Law of Thermodynamics for Quantum Damped Oscillators Rami Ahmad El-Nabulsi 1 Received: 6 August 2020 / Accepted: 3 October 2020/ # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
In this study, we have generalized the fractional action integral by using the SaigoMaeda fractional operators defined in terms of the Appell hypergeometric function of two variables , F3(a, a′, β, β′; γ; z, ζ) with complex parameters. We have derived the associated Euler-Lagrange equation and we have studied the harmonic oscillator problem. We have proved that a PT -symmetric quantum-mechanical Hamiltonian characterized by real and discrete spectra is obtained although the system is characterized by complex trajectories. The associated thermodynamical properties were discussed and it was revealed the entropy of the quantum system decreases with time toward an asymptotically positive value similar to what is observed in quantum Maxwell demon. Keywords Saigo-Maeda fractional operators . Appell hypergeometric function . Quantum damped oscillator . Violation of the 2nd law of thermodynamics AMS Subject Classification 49 K05 . 49S05
1 Introduction The Fractional Calculus of Variations (FCoV) has received recently a considerable interest due to its large implications in several fields of sciences and engineering including material sciences, quantum mechanics, physical kinetics and transport theory among others [1, 2]. In particular, FCoV is successfully to physical processes subject to dissipations [3–10] allowing
* Rami Ahmad El-Nabulsi [email protected]
1
Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671 Athens, Greece
International Journal of Theoretical Physics
to apply the fractional variational methods to real dynamical systems. Within the fractional approach, external forces emerged naturally in the dynamical equations through the associated fractional Euler-Lagrange equation [11–14]. Several different approaches to FCoV generalizing the least action principle were introduced in literature. Results include theoretical mathematical problems depending on dissimilar types fractional derivatives and integrals operators, e.g. Caputo, Riemann-Liouville, Hadamard, Riesz, Erdelyi-Kober, etc. [15, 16]. One successful approach is known as the fractional actionlike variational approach based on the Riemann-Liouville fractional integrals which is the most extensively used integral of fractional order 0 < α ≤ 1 via an integral transform defined in Lebesgue integrable space by [4, 13]: aI t
α
f ðtÞ ¼
1 t ∫ f ðτ Þðt−τ Þα−1 dτ Γ ðαÞ a
ð1Þ
α
f ðt Þ ¼
1 b ∫ f ðτ Þðτ−tÞα−1 dτ Γ ðαÞ t
ð2Þ
tIb
which are respectively the left and right fractional integrals where the former is valid for t > a ∞ and the latter is valid for t < b. Γ ðαÞ ¼ ∫0 tα−1 expð−tÞ dt is the Euler gamma function. The fractional action is give
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