Part of the Lecture Notes in Mathematics book series (LNM, volume 1954)


The term potential theory comes from 19th century physics, where the fundamental forces like gravity or electrostatic forces were described as the gradients of potentials, i.e. functions which satisfy the Laplace equation. Hence potential theory was the study of solutions of the Laplace equation. Nowadays the fundamental forces in physics are described by systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations, and the Laplace equation arises only as a limiting case. Nevertheless, the Laplace equation is still used in applications in many areas of physics and engineering like heat conduction and electrostatics. And the term “potential theory” has survived as a convenient label for the theory of functions satisfying the Laplace equation, i.e. so-called harmonic functions.


Laplace Equation Potential Theory Dirichlet Form Quantum Walk Quantum Probability 
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© Springer-Verlag Berlin Heidelberg 2008

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