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Harper Equation

The Harper equation is the difference equation

u_(n+1)+u_(n-1)+2cos[2pi(nalpha+theta)]u_n=Eu_n.

It is the critical-coupling case of the almost Mathieu operator eigenvalue equation and models single-band electron motion on a two-dimensional lattice in a perpendicular magnetic field. Plotting its operator spectrum against the magnetic flux parameter alpha produces Hofstadter's butterfly.

See also

Almost Mathieu Operator, Difference Equation, Hofstadter's Butterfly, Operator Spectrum

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References

Harper, P. G. "Single Band Motion of Conduction Electrons in a Uniform Magnetic Field." Proc. Phys. Soc. A 68, 874-878, 1955. https://doi.org/10.1088/0370-1298/68/10/304.

Cite this as:

Weisstein, Eric W. "Harper Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarperEquation.html

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