Quasiclassical asymptotics and coherent states for bounded discrete spectra

Journal Article · Wed Dec 15 04:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3503775· OSTI ID:21501208

Gorska, K ^[1]; Penson, K A ^[2]; Horzela, A; Blasiak, P ^[3]; Duchamp, G H. E. ^[4]; Solomon, A I ^[2]

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1. Nicolaus Copernicus University, Institute of Physics, ul. Grudziadzka 5/7, PL 87-100 Torun (Poland)
2. Laboratoire de Physique Theorique de la Matiere Condensee, Universite Pierre et Marie Curie, CNRS UMR 7600, Tour 13 - 5ieme et., B.C. 121, 4 pl. Jussieu, F 75252 Paris Cedex 05 (France)
3. H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31342 Krakow (Poland)
4. Universite Paris XIII, LIPN, Institut Galilee, CNRS UMR 7030, 99 Av. J.-B. Clement, F 93430 Villetaneuse (France)

We consider discrete spectra of bound states for nonrelativistic motion in attractive potentials V{sub {sigma}}(x)=-|V{sub 0}| |x|{sup -}{sigma}, 0<{sigma}{<=}2. For these potentials the quasiclassical approximation for n{yields}{infinity} predicts quantized energy levels e{sub {sigma}}(n) of a bounded spectrum varying as e{sub {sigma}}(n){approx}-n{sup -}2{sigma}/(2-{sigma}). We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For {sigma} in the range 0 < {sigma}{<=} 2/3 we present exact implementations of such states for the parametrization {sigma}= 2(k-l)/(3k-l) with k and l positive integers satisfying k>l.

• OSTI ID: 21501208
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 51; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANNIHILATION OPERATORS
APPROXIMATIONS
ASYMPTOTIC SOLUTIONS
CALCULATION METHODS
EIGENSTATES
ENERGY LEVELS
EXACT SOLUTIONS
FUNCTIONS
INTEGRAL TRANSFORMATIONS
MATHEMATICAL OPERATORS
MATHEMATICAL SOLUTIONS
MELLIN TRANSFORM
QUANTUM OPERATORS
QUANTUM STATES
SPECTRA
TRANSFORMATIONS
WAVE FUNCTIONS