Abstract
The N-dimensional quantum Hamiltonianis shown to be exactly solvable for any real positive value of the parameter η. Algebraically, this Hamiltonian system can be regarded as a new maximally superintegrable η-deformation of the N-dimensional Kepler–Coulomb Hamiltonian while, from a geometric viewpoint, this superintegrable Hamiltonian can be interpreted as a system on an N-dimensional Riemannian space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation parameter η and of the coupling constant k.
Highlights
Let us consider the N -dimensional (N D) classical Hamiltonian given by |q|p2 k Hη(q, p) = Tη(q, p) + Uη (q) 2(η |q|) − η
A detailed analysis of the different possible quantization prescriptions together with a proof of their equivalence through gauge transformations will be presented in a forthcoming paper [21]. One of this prescriptions consists in the so called ‘direct’ or Schrodinger quantization [15], under which the quantum Hamiltonian Hη keeps the maximal superintegrability property and is endowed with (2N −1) algebraically independent operators that commute with Hη
Spectrum and eigenfunctions In view of the effective potential introduced in (10), one should expect that the quantum Hamiltonian (11) should have both a discrete and a continuous spectrum, and this is the case
Summary
Theorem [22] to 3D conformally flat Riemannian spaces, providing a more general notion of KC and harmonic oscillator potentials [23, 24, 25]. According to the above definitions and by considering the Taub-NUT metric (4), with conformal factor η f (r) = 1 + , r it is straightforward to obtain that the corresponding intrinsic potentials read η
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