Distribution of eigenvalues in the presence of higher order turning points

Author:
Anthony Leung

Journal:
Trans. Amer. Math. Soc. **229** (1977), 111-135

MSC:
Primary 34B25; Secondary 34E20

MathSciNet review:
0447699

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article is concerned with the eigenvalue problem , where is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on , 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.

**[1]**M. A. Evgrafov and M. V. Fedorjuk,*Asymptotic behavior of solutions of the equation 𝑤′′(𝑧)-𝑝(𝑧,𝜆)𝑤(𝑧)=0 as 𝜆→∞ in the complex 𝑧-plane*, Uspehi Mat. Nauk**21**(1966), no. 1 (127), 3–50 (Russian). MR**0209562****[2]**Anthony Wing Kwok Leung,*Connection formulas for asymptotic solutions of second order turning points in unbounded domains*, SIAM J. Math. Anal.**4**(1973), 89–103. MR**0333382****[3]**Anthony Leung,*Lateral connections for asymptotic solutions around higher order turning points*, J. Math. Anal. Appl.**50**(1975), 560–578. MR**0372356****[4]**Yasutaka Sibuya,*Subdominant solutions of the differential eqauation 𝑦′′-𝜆²(𝑥-𝑎₁)(𝑥-𝑎₂)\cdots(𝑥-𝑎_{𝑚})𝑦=0*, Acta Math.**119**(1967), 235–272. MR**0224930****[5]**Wolfgang Wasow,*Asymptotic expansions for ordinary differential equations*, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0203188****[6]**Laurence Weinberg,*The asymptotic distribution of eigenvalues for the boundary value problem 𝑦′′(𝑥)-𝜆²𝑝(𝑥)𝑦(𝑥)=0,𝑦∈𝐿₂(-∞,+∞)*, SIAM J. Math. Anal.**2**(1971), 546–566. MR**0306656****[7]**Anthony Wing Kwok Leung,*Connection formulas for asymptotic solutions of second order turning points in unbounded domains*, SIAM J. Math. Anal.**4**(1973), 89–103. MR**0333382**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34B25,
34E20

Retrieve articles in all journals with MSC: 34B25, 34E20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0447699-3

Keywords:
Linear differential equations,
boundary value problem,
eigenvalues,
turning points,
connection formulas

Article copyright:
© Copyright 1977
American Mathematical Society