Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, Schrodinger's, Einstein's and Newton's equations, and others. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases, e.g., the case of quartic oscillators, these methods do not work. New geometric methods, which have the advantage of providing...

A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.

With each methodology treated in its own chapter, this monograph is a thorough exploration of several theories that can be used to find explicit formulas for heat kernels for both elliptic and sub-elliptic operators. The authors show how to find heat kernels for classical operators by employing a number of different methods. Some of these methods come from stochastic processes, others from quantum physics, and yet others are purely mathematical. Depending on the symmetry, geometry and ellipticity, some methods are more suited for certain operators rather than others.

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