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Atkinson-Mingarelli Theorem

The Atkinson-Mingarelli theorem gives the asymptotic distribution of the positive and negative eigenvalues of an indefinite weighted Sturm-Liouville equation

-d/(dx)(p(x)(dy)/(dx))+q(x)y=lambdaw(x)y.
(1)

Suppose 1/p, q, and w are real-valued functions that are integrable on a closed interval [a,b], separated boundary conditions are imposed, and p has only finitely many sign changes. Define (w/p)_+=max(w/p,0) and (w/p)_-=max(-w/p,0), and suppose neither is identically zero. Then there are sequences of positive and negative real eigenvalues whose asymptotic behavior is

lambda_n^+∼(n^2pi^2)/([int_a^bsqrt((w/p)_+(x))dx]^2)
(2)
lambda_n^-∼(-n^2pi^2)/([int_a^bsqrt((w/p)_-(x))dx]^2),
(3)

as n->infty. Thus the positive and negative portions of w/p determine the two spectral branches separately.

See also

Boundary Value Problem, Eigenvalue, Sturm-Liouville Equation

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References

Mingarelli, A. B. and Atkinson, F. V. "Asymptotics of the Number of Zeros and of the Eigenvalues of General Weighted Sturm-Liouville Problems." J. Reine Angew. Math. 375-376, 380-393, 1987. https://doi.org/10.1515/crll.1987.375-376.380.

Cite this as:

Weisstein, Eric W. "Atkinson-Mingarelli Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Atkinson-MingarelliTheorem.html

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