Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound sta
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AIRY FUNCTIONS AND TRANSITION BETWEEN SEMICLASSICAL AND HARMONIC OSCILLATOR APPROXIMATIONS FOR ONE-DIMENSIONAL BOUND STATES A. Yu. Anikin,∗ S. Yu. Dobrokhotov,∗ and A. V. Tsvetkova∗
We consider the one-dimensional Schr¨ odinger operator with a semiclassical small parameter h. We show that the “global” asymptotic form of its bound states in terms of the Airy function “works” not only for excited states n ∼ 1/h but also for semi-excited states n ∼ 1/hα , α > 0, and, moreover, n starts at n = 2 or even n = 1 in examples. We also prove that the closeness of such an asymptotic form to the eigenfunction of the harmonic oscillator approximation.
Keywords: bound state, Schr¨ odinger operator, semiclassical approximation, asymptotics, eigenfunction, harmonic oscillator, Airy function DOI: 10.1134/S0040577920080024
1. Introduction We consider the well-known question about the asymptotic behavior of solutions of the spectral problem as h → 0 for the one-dimensional Schr¨ odinger equation 2
≡ − h Ψ (x) + V (x)Ψ = EΨ, HΨ 2
ΨL2 (R) = 1.
(1)
We assume that the function V (x) is smooth and bounded below and grows at infinity not faster than has a discrete spectrum in this case. For simplicity, we assume that V (0) = 0 polynomially. The operator H is the global nondegenerate minimum of V (x). We also assume that En and Ψn are sets of eigenvalues and normalized eigenfunctions of (1) such that En ∈ (0, E) and the En are arranged in ascending order. The quantity E > 0 is here sufficiently small but independent of h. The semiclassical approximation (see [1], [2]) is used to describe the asymptotic behavior of the numbers En and the functions Ψn (x) corresponding to the order numbers n ∼ 1/h, which are called the upper levels (or excited states). We let H(p, x) = p2 /2 + V (x) denote the classical Hamiltonian corresponding to Eq. (1) and let Λ(E) = {(p, x) : H = E} denote the closed curves that lie on the energy level E ∈ (0, E) (they are trajectories of the Hamiltonian system with the Hamiltonian H). The asymptotic form of En is then determined with an error of O(h2 ) from the Bohr–Sommerfeld quantization rule 1 π
1 +n , 2(E − V (x)) dx = h 2
x+ (E)
x− (E)
n ∈ Z,
(2)
∗
Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia, e-mail: [email protected]. This research is supported by the Russian Foundation for Basic Research (Grant No. 18-31-00273) and was also supported by the Federal Target Program (No. AAAA-A17-117021310377-1). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 171–180, August, 2020. Received March 23, 2020. Revised March 23, 2020. Accepted March 27, 2020. 984
c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-0984
where x− (E) < x+ (E) are roots of the equation E − V (x) = 0 (they are called turning points). The asymptotic behavior of eigenfunctions is determined by the “quantized” curves Λn = Λ(En ) and is given by the expression ψn (x) = KΛn 1/KΛn 1, where KΛn is the Maslov canonical operator on these closed trajectories. The standard semiclas
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