Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$ - INSPIRE

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Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$

C. Quesne
(
- Brussels U., PNTPM
)

Nov 23, 2009

11 pages

Published in:

J.Phys.A 43 (2010) 082001

e-Print:

0911.4404 [math-ph]

DOI:

10.1088/1751-8113/43/8/082001

Report number:

ULB-229-CQ-09-4

View in:

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Abstract: (arXiv)

In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of

k

, some novel exactly solvable and integrable quantum Hamiltonian

H_k

on a plane is superintegrable and that the additional integral of motion is a

2k

th-order differential operator

Y_{2k}

. Here we demonstrate the conjecture for the infinite family of Hamiltonians

H_k

with odd

k \ge 3

, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some

D_{2k}

-extended and invariant Hamiltonian \chh_k, which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a

D_{2k}

-invariant integral of motion \cyy_{2k}, from which

Y_{2k}

can be obtained by projection in the

D_{2k}

identity representation space.

Note:

14 pages, no figure; change of title + important addition to sect. 4 + 2 more references + minor modifications; accepted by JPA as an FTC

03.65.Fd
operator: differential
Hamiltonian
integrability
oscillator

References(15)

Figures(0)

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