Summary
In this paper we study a mixed equation,i.e. a finite-difference one with respect to time and a differential one with respect to a continuous variablex, which reduces to the Schrödinger equation for the linear harmonic oscillator in the proper continuum limit. We construct its solution explicitly by a suitable algorithm, which is also shown to have applications in the theory of Hermite polynomials.
Riassunto
In questo lavoro si studia un’equazione mista, alle differenze finite rispetto al tempo e differenziale in una variabile continuax, che si riduce all’equazione di Schrödinger per l’oscillatore lineare armonico nell’appropriato limite continuo. Se ne costruisce esplicitamente la soluzione, con un conveniente algoritmo, del quale si illustrano poi alcune applicazioni alla teoria dei polinomi di Hermite.
Резюме
В этой работе мы исследуем смешанное уравнение, т.е. уравнение в конечных разностях по времени и дифференциальное по непрерывной переменнойx, которое сводится к уравнению Шредингера для линейного гармонического осцилятора в соответствующем непрерывном пределе. Используя соответствующий алгоритм, мы конструируем решение этого уравнения, которое применимо в теории полиномов Эрмита.
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References
F. Casagrande andE. Montaldi:Nuovo Cimento,40 A, 369 (1977).
See alsoA. J. Phares:Journ. Math. Phys.,18, 1838 (1977), where a general formalism is developed.
A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.2 (New York, N. Y., 1954), p. 183.
A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.2 (New York, N. Y., 1954), p. 193 and 189.
Y. L. Luke:The Special Functions and Their Approximations, Vol.1 (New York, N. Y., 1969), p. 18.
E. Montaldi:Lett. Nuovo Cimento,15, 139 (1976). See alsoA. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.3 (New York, N. Y., 1954), p. 254.
A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.2 (New York, N. Y., 1954), p. 194.
A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.2 (New York, N. Y., 1954), p. 176.
E. Montaldi:Lett. Nuovo Cimento,15, 139 (1976). See alsoA. Erdélyi, W. Magnus, F. Oberhettinger andF. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.3 (New York, N. Y., 1954).
A. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.1 (New York, N. Y., 1954), p. 82.
seeA. Erdélyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Bateman Manuscript Project, Higher Transcendental Functions, Vol.2 (New York, N. Y., 1954), p. 194.
E. Montaldi:Lett. Nuovo Cimento,15, 139 (1976), eq. (24).
L. Infeld andT. E. Hull:Rev. Mod. Phys.,23, 21 (1951).
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Casagrande, F., Montaldi, E. On the discrete-time Schrödinger equation for the linear harmonic oscillator. Nuov Cim A 44, 453–464 (1978). https://doi.org/10.1007/BF02896334
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DOI: https://doi.org/10.1007/BF02896334