The Schroedinger operator as a generalized Laplacian

Details The Schroedinger operator as a generalized Laplacian The Schroedinger operator as a generalized Laplacian

2007-11-18

arxiv.org/abs/0711.2777v2

J. Phys. A: Math. Theor. 41 (2008), 145204

Physics

Mathematical Physics

19 pages, a remark, an example and references added - the version to appear in J. Phys. A: Math. and Theor

Scientific paper

10.1088/1751-8113/41/14/145204

The Schroedinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schroedinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -- the Schroedinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U(1)-principal bundle with an analogous list of trivializations -- the Schroedinger principal bundle. For the Schroedinger principal bundle a natural differential calculus for `wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudo-Riemannian metric. The free Schroedinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schroedinger principal bundle. The presented framework is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory.

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