The Schrödinger operator as a generalized Laplacian

The Schrödinger 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 Schrödinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle – the Schrödinger 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 Schrödinger principal bundle. If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over space-time, but as there is no canonical trivialization (inertial frame), these sections interpreted as ‘wave-functions’ cannot be viewed as actual functions on the space-time. In this approach the change of an observer results not only in the change of actual coordinates in the space-time but also in a change of the phase of wave functions. For the Schrödinger 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 pseudoRiemannian metric. The free Schrödinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame – which is supported by producing right physics predictions – and it 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. MSC 2000: 35J10, 70G45.