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Eigenform

Given a differential operator D on the space of differential forms, an eigenform is a form alpha such that

Dalpha=lambdaalpha
(1)

for some constant lambda. For example, on the torus, the Dirac operator D=-i(d+d^*) acts on the form

beta=3e^(i(3x+4y))+5e^(i(3x+4y))dx-4e^(i(3x+4y))dx ^ dy,
(2)

giving

Dbeta=15e^(i(3x+4y))+25e^(i(3x+4y))dx-20e^(i(3x+4y))dx ^ dy,
(3)

i.e., Dbeta=5beta.

See also

Eigen Decomposition, Laplacian, Operator Spectrum

This entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Eigenform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Eigenform.html

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