In this paper the geometric theory of separation of variables for the time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This task is performed without complexifying the manifold but just by considering complex-valued functions on it. The simple formalism introduced in the paper allows us to extend in a very natural way the classical results on separation of variables (including Levi-Civita criterion and Stäckel-Eisenhart theory) to the complex case. Only orthogonal variables are considered.
REFERENCES
1.
Benenti
, S.
, “Intrinsic characterization of variable separation in the Hamilton-Jacobi equation
,” J. Math. Phys.
38
, 6578
–6602
(1997
).2.
Benenti
, S.
, Chanu
, C.
, and Rastelli
, G.
, “Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds
,” J. Math. Phys.
42
, 2065
–2091
(2001
).3.
Benenti
, S.
, Chanu
, C.
, and Rastelli
, G.
, “Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation I and II
,” J. Math. Phys.
43
, 5183
–5253
(2002
).4.
Chanu
, C.
, Degiovanni
, L.
, and McLenaghan
, R. G.
, “Geometrical classification of Killing tensors on bidimensional flat manifolds
,” J. Math. Phys.
47
, 073506
(2006
).5.
Chanu
, C.
, and Rastelli
, G.
, “Eigenvalues of Killing Tensors and separable webs on Riemannian and pseudo-Riemannian manifolds
,” SIGMA Symmetry Integrability Geom. Methods Appl.
3
, Paper 021, 21
(2007
).6.
Degiovanni
, L.
, and Rastelli
, G.
, “Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space
,” Int. J. Geom. Methods Mod. Phys.
(to be published) nlin/0612051.7.
Dubrovin
, B. A.
, Fomenko
, A. T.
, and Novikov
, S. P.
, Modern Geometry—Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields
, 2nd ed., Graduate Texts in Mathematics
, Vol. 93
(Springer-Verlag
, Berlin
, 1992
).8.
Eisenhart
, L. P.
, “Separable systems of Stackel
,” Ann. Math.
35
, 284
–305
(1934
).9.
Kalnins
, E. G.
, and Miller
, Jr., W.
, “Separable coordinates for three-dimensional complex Riemannian spaces
,” J. Diff. Geom.
14
, 221
–236
(1979
).10.
Kalnins
, E. G.
, and Miller
, Jr., W.
, “Nonorthogonal -separable coordinates for four-dimensional complex Riemannian spaces
,” J. Math. Phys.
22
, 42
–50
(1981
).11.
Hörmander
, L.
, An Introduction to Complex Analysis in Several Variables
, North-Holland Mathematical Library
Vol. 7
, 3rd ed. (North-Holland
, Amsterdam
, 1990
).12.
Levi-Civita
, T.
, “Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili
,” Math. Ann.
59
, 3383
–3397
(1904
).13.
Pucacco
, G.
, and Rosquist
, K.
, Proceeding of SPT 2002: Symmetry and Perturbation Theory, Cala Gonone
(World Scientific
, Singapore
, 2002
), pp. 196
–209
.© 2007 American Institute of Physics.
2007
American Institute of Physics
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