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How is the Diamagnetic inequality born? Why is it call this name?

Diamagnetic inequality: $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.

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Why is it called this name?

Diamagnetism is a quantum effect in which material under an applied external magnetic field generate their own "opposite" magnetic field. The most prominent example happens in superconductors.

A consequence of this is that matter under an applied external magnetic field "gain" energy (or rather, removing magnetic fields decreases kinetic energy). See, for example, this paper. Now, the quantum mechanical description of the momentum operator is just $i\nabla$, and the associated (free) Hamiltonian is $-\triangle = (i\nabla)^2$. The mathematical description of a quantum mechanical system under an external magnetic field uses the gauge potential $A$ to capture the magnetic field, and postulates that the Hamiltonian is $H = (i\nabla + A)^2$. And hence results which bound the lower energy $i\nabla$ by the higher energy $i\nabla + A$ became known as "diamagnetic inequalities". See also this mathscinet review where this is briefly discussed.

Note that originally inequalities of the type are (among many) called Kato's inequalities. Now they are called diamagnetic inequalities because it conveniently reminds us of the physics.

For how to derive the inequality and some more historical and practical notes, see sections 7.19 - 7.22 in Lieb and Loss, Analysis.

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