ISSN 1063-7788, Physics of Atomic Nuclei, 2007, Vol. 70, No. 3, pp. 513–519. c  Pleiades Publishing, Ltd., 2007. SECOND INTERNATIONAL WORKSHOP ON SUPERINTEGRABLE SYSTEMS IN CLASSICAL AND QUANTUM MECHANICS Theory Noncommuting Limits of Oscillator Wave Functions * J. Daboul 1)** , G. S. Pogosyan 2)*** , and K. B. Wolf 3)**** Received May 16, 2006 Abstract—Quantum harmonic oscillators with spring constants k> 0 plus constant forces f exhibit rescaled and displaced Hermite–Gaussian wave functions, and discrete, lower bound spectra. We examine their limits when (k,f ) → (0, 0) along two different paths. When f → 0 and then k → 0, the contraction is standard: the system becomes free with a double continuous, positive spectrum, and the wave functions limit to plane waves of definite parity. On the other hand, when k → 0 first, the contraction path passes through the free-fall system, with a continuous, nondegenerate, unbounded spectrum and displaced Airy wave functions, while parity is lost. The subsequent f → 0 limit of the nonstandard path shows the dc hysteresis phenomenon of noncommuting contractions: the lost parity reappears as an infinitely oscillating superposition of the two limiting solutions that are related by the symmetry. PACS numbers: 03.65.-w DOI: 10.1134/S1063778807030106 1. INTRODUCTION: OSCILLATORS AND FORCES The one-dimensional harmonic oscillator is a me- chanical system whose restitution force −kx is pro- portional to the spring constant k> 0 and opposed to the separation x between a mass point and the oscillator center. When this system is subjected to a constant external force −f (f> 0, such as a gravita- tional field in the direction of the negative x axis), it is characterized by the Hamiltonian operator H (k,f ) (ˆ x, ˆ p) := 1 2 ˆ p 2 + 1 2 k ˆ x 2 + f ˆ x (1) for unit mass, where ˆ x and ˆ p denote the Schr ¨ odinger operators of position and momentum, with units cho- sen so that [ˆ x, ˆ p]= i1 [1]. The pair of parameters (f,k) thus provide a plane to study the contraction limits to the free quantum particle at the point (0, 0) by taking various paths. The purpose of this paper is to analyze a case where there are two inequivalent paths to reach the ∗ The text was submitted by the authors in English. 1) On sabatical leave from Physics Department, Ben Gurion University of the Negev, Beer Sheva, Israel. 2) Departamento de Matem ´ aticas, Universidad de Guadala- jara, M ´ exico; on leave from the Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Rus- sia. 3) Instituto de Ciencias F ´ ısicas, Universidad Nacional Aut ´ onoma de M ´ exico, Cuernavaca, M ´ exico. ** E-mail: daboul@bgu.ac.il *** E-mail: pogosyan@thsun1.jinr.ru **** E-mail: bwolf@fis.unam.mx free-particle limit, as shown in Fig. 1. There is the standard path, where first f is turned off so the oscillator H (k,0) is centered on the origin, followed by the limit k → 0 to the free H (0,0) . And there is the nonstandard path, which first turns off k, so that the system H (0,f ) is that of free fall, and then lets the force f vanish [2]. The problem posed by this non- commutation of deformation and contraction (for N - dimensional systems), called “dc hysteresis,” was fol- lowed through the symmetry algebras of the Hamil- tonians on the paths of Fig. 1. Here, we examine the case of N =1-dimensional quadratic systems, where the symmetry group of (1) is parity under reflections across the oscillator center, which exchanges the two turning points of the harmonic motion for every en- ergy; this continues being a symmetry in the free limit. In the free-fall system, however, this symmetry under reflection is lost; there is only one point of return. In the subsequent free limit, the symmetry cannot be fully recovered; but—is it lost? Here, we examine the phenomenon of dc hysteresis in the spectra and the wave functions of the oscillator, free-fall, and free systems, with the case of N =1 dimension show- ing this phenomenon clearly through the asymptotic properties of special functions. In Section 2, we formalize the proposed limits classically, and in Section 3, we write the Hermite functions of the displaced harmonic oscillator in a form suitable for the proposed limits [3]. These are performed in Section 4 to the free particle, where Hermite functions limit to trigonometric functions, and in Section 5 to the free-fall system, where the limit is to Airy functions. Section 6 examines the 513