Quantum integrability of the geodesic flow for c-projectively equivalent metrics

Abstract

Two (pseudo-) Riemannian metrics are called projectively equivalent, if they possess the same geodesics (considered as unparametrized curves). C-projective eqivalence is a natural translation of projective equivalence to Kahler manifolds: A regular curve on a Kahler manifold (M, g, J) is called J-planar, if the acceleration lies within the span of the tangent vector and J applied to the tangent vector. Two Kahler metrics on a complex manifold of real dimension larger than four are called c-projectively equivalent, if every J-planar curve of one metric is also J-planar with respect to the other metric. In the main part we will tackle the following questions: Firstly: Do the integrals of the geodesic flow commute as quantum operators? This could successfully be answered with yes. Secondly: Is a generalization to natural Hamiltonian systems possible? This means: is it possible to add potentials to the integrals of the geodesic flow such that the resulting functions still commute with respect to the Poisson bracket? Furthermore do the assigned quantum operators still commute? The answer is yes. The admissible potentials are the same at the level of classical integrability as well as at the quantum level. For certain cases all admissible potentials could be described. For other cases a family of admissible potentials will be given, but there may be more admissible potentials. Thirdly: For Kahler metrics that possess c-projectively equivalent metrics the separation of variables for Schrodinger’s equation was investigated: We showed that the search for simultaneous eigenfunctions of the previously constructed quantum operators can be reduced to differential equations in lower dimension, in the case of maximal integrability a reduction to ordinary differential equations is possible.