Event
Alexander Turbiner, UNAM, M茅xico
Tuesday, October 3, 2017 15:30to16:30
Room 4336, Pav. Andr茅-Aisenstadt, 2920, ch. de la Tour, Montreal, QC, CA
The quantum n-body problem in dimension d ≥ n-1: the ground state.
We employ generalized Euler coordinates for the n body system in d 鈮� n 鈭� 1 dimensional space, which consist of the centre-of-mass vector, relative (mutual) distances r_ij and angles as remaining coordinates. We prove that the kinetic energy of the quantum n-body problem for d 鈮� n 鈭� 1 can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator D_1 which depends on relative distances alone and (iii) the differential operator D_2 which annihilates any angle-independent function. The operator D_1 has a large reflection symmetry group (a direct sum of n(n-1)/2 copies of Z_2) and in 蟻_ij = r_{ij}^2 variables is the algebraic operator with hidden algebra sl(n(n鈭�1)/2 + 1, R). Thus, it is the Hamiltonian of quantum Euler-Arnold sl(n(n鈭�1)/2 + 1, R) top in a constant magnetic field. It is conjectured that for any n similarity-transformed D_1 is the Laplace-Beltrami operator plus (effective) potential, thus, it describes a n(n鈭�1)/2-dimensional quantum particle in curved space, it was verified for n = 2, 3, 4. After de-quantization similarity-transformed D_1 becomes the Hamiltonian of the classical top with variable tensor of inertia in external potential. Work done with W. Miller, Jr. and A. Escobar-Ruiz.
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CRM, UdeM, Pavillon Andr茅-Aisenstadt, 2920, ch. de la Tour, salle 4336
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