The Asymptotic Distribution of Eigenvalues for the Boundary Value Problem $y''(x) - \lambda ^2 p(x)y(x) = 0,y \in L_2 ( - \infty , + \infty )$

Weinberg, Laurence

doi:10.1137/0502056

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1972

1979

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“…Problems of this type are of interest in the study of particle scattering and wave mechanics. Sibuya [4], Weinberg [6], and Evgrafov and Fedoryuk [1] have studied the asymptotic distribution of large positive eigenvalues for problems (1.1), (1.2). However, essentially they all assumed ihatp(x) can only have pairs of first order zeroes on the real axis.…”

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confidence: 99%

Distribution of eigenvalues in the presence of higher order turning points

Leung¹

1977

Trans. Amer. Math. Soc.

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Abstract. This article is concerned with the eigenvalue problem u"(x) -A^j(jc)u(jc) = 0, u(x) E ¿2(-oo, oo), where p(x) is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on p(x), approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.

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confidence: 99%