Ein Verfahren höherer Konvergenzordnung zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung

Summary

The characteristic exponent ν of Mathieu's differential equation

$$y''(x) + 4(\lambda + 2t\cos 2x)y(x) = 0$$

satisfies the relation

$$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \sin ^2 (\pi \sqrt \lambda )\det S^{(0)} \det C^{(0)} ,$$

if λ≠n ² (n∈ℕ), and an analogous equation for λ≠(n+1/2)², whereS ⁽⁰⁾ andC ⁽⁰⁾ are certain infinite tridiagonal matrices. We calculate the determinants ofS ⁽⁰⁾=(σ _n,m ^∞₀ andC ⁽⁰⁾ using

$$\det S^{(0)} = \prod\limits_{n = 0}^\infty {(1 - \beta _n ){\text{ }}\det B,{\text{ }}B = \left( {\frac{{\sigma _{n,m} }}{{1 - \beta _n }}} \right)_0^\infty ,}$$

where the constants (1−β _n ) are chosen in such way that the infinite product may be evaluated by trigonometric functions and the finite determinants detB _N converge like a series with termsO(N ⁻¹²).