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Mathieu Operator

The Mathieu operator is the differential operator with cosine potential

(L_qy)(x)=-y^('')(x)+2qcos(2x)y(x).

Its eigenvalue equation L_qy=ay is the Mathieu differential equation y^('')(x)+[a-2qcos(2x)]y(x)=0 (Anahtarci and Djakov 2012, NIST DLMF). On the interval [0,pi], the periodic conditions y(pi)=y(0), y^'(pi)=y^'(0) and the antiperiodic conditions y(pi)=-y(0), y^'(pi)=-y^'(0) give the two standard boundary value problems.

The almost Mathieu operator is a discrete cosine-potential counterpart, acting on doubly infinite sequences instead of functions of a continuous variable.

See also

Almost Mathieu Operator, Mathieu Differential Equation, Mathieu Function

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References

Anahtarci, B. and Djakov, P. "Refined Asymptotics of the Spectral Gap for the Mathieu Operator." J. Math. Anal. Appl. 396, 243-255, 2012. https://doi.org/10.1016/j.jmaa.2012.06.019.National Institute of Standards and Technology. "Definitions and Basic Properties." §28.2 in Digital Library of Mathematical Functions. https://dlmf.nist.gov/28.2.

Cite this as:

Weisstein, Eric W. "Mathieu Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MathieuOperator.html

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