Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model |
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| Institution: | 1. Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan;2. Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan;3. I.E.Tamm Theory Department, Lebedev Physics Institute, Leninsky prospect, 53, Moscow 119991, Russia;4. ITEP, B. Cheremushkinskaya, 25, Moscow 117259, Russia;5. Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build.1, Moscow 127051, Russia;1. National Research University Higher School of Economics, Russian Federation;2. Skolkovo Institute of Science and Technology, 143026 Moscow, Russian Federation;3. Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russian Federation;4. ITEP, B. Cheremushkinskaya 25, Moscow 117218, Russian Federation;5. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, Moscow Region 141700, Russian Federation;1. Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy;2. INFN, Sezione di Bari, I-70126 Bari, Italy;3. Dipartimento di Scienze Fisiche and MECENAS, Università di Napoli “Federico II”, I-80126 Napoli, Italy;4. INFN, Sezione di Napoli, I-80126 Napoli, Italy;5. Raman Research Institute, 560080 Bangalore, India |
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| Abstract: | Using a similarity transformation that maps the Calogero model into N decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions.The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the BN-Calogero model. |
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