Geometric Analysis on the Heisenberg Group and Its Generalizations

Description

The book deals with the study of subRiemannian manifolds which have the Heisenberg principle built in. This brings the hope that Heisenberg manifolds (step 2 subRiemannain manifolds) will play a role in Quantum Mechanics in the future, similar to the role played by the Riemannian manifolds in Classical Mechanics.

The subRiemannian geodesics behavior, which is very different than in the Riemannian case, plays an important role in finding heat kernels and propagators for the Shrodinger’s equation, as well as in finding fundamental solutions for subelliptic operators using a very ingenious geometric method involving complex modified action.

One of the novelties of this book is to introduce the complex Hamiltonian mechanics techniques and apply them in describing the fundamental solutions and heat propagators. The fact that the propagator depends on each subRiemannian distance is an expected fact in Quantum Mechanics. The fact that each path has a contribution to the propagator led Feynman to introduce the path integrals. In this work we can manage without them, just working in the proper geometric framework.

This work is a text for a course or seminars designed for graduate students interested in the most recent developments in the subRiemannian manifolds and sub-elliptic operators theory. It is useful for both pure and applied mathematicians and theoretical physicists working in the quantum mechanics area.

This volume is part of the AMS/IP Studies in Advanced Mathematics book series.

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