TY - JOUR
T1 - Lie theory and separation of variables. 4. The groups SO(2,1) and SO(3)
AU - Kalnins, E. G.
AU - Miller, Willard
PY - 1973
Y1 - 1973
N2 - Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x02 - x1 2-x22 = 1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one-variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well-known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lamé functions. We show that there is virtually a perfect correspondence between the known theory of the Lamé functions and the representation theory of SO (2,1) and SO (3).
AB - Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x02 - x1 2-x22 = 1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one-variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well-known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lamé functions. We show that there is virtually a perfect correspondence between the known theory of the Lamé functions and the representation theory of SO (2,1) and SO (3).
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U2 - 10.1063/1.1666805
DO - 10.1063/1.1666805
M3 - Article
AN - SCOPUS:36849112731
SN - 0022-2488
VL - 15
SP - 1263
EP - 1274
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 8
ER -