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By W. N. Everitt, B. D. Sleeman

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Extra info for Ordinary and Partial Differential Equations: Proceedings of the Seventh Conference Held at Dundee, Scotland, March 29 - April 2, 1982

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Soc. Edin. A 74, Some Limit Circle Eigenvalue Problems and Asymptotic Formulae for Eigenvalues' by F V Atkinson and C T Fulton §i Introduction Asymptotic properties of eigenvalues of the second order equation -y" + qy = %y on the half line [0,~) with a regular boundary condition at obtained by a number of authors under various conditions on q(x) ÷ + ~ as x-~ ~ We mention, in particular, x = 0 q(x) have been in the case when the work of Titchmarsh [24,26, 27], McLeod and Titchmarsh [21], MeLeod [19,20], Giertz [9], K J~rgens [17], Levitan and Sargsjan [18, Chap.

Transl. (2),103, 19-102 (1974), (Supplement II to reference no. 1 above, "Mir", Moscow, 1968). 15. M. Levitan, On the asymptotic behaviour of the spectral function of a selfadjoint second-order differential equation, Izv. Akad. Nauk SSSR Ser. Mat. 16, 325-352 (1952), also Amer. Math. Soc. (2) lO1, 192-221 (1973). 16. M. Levitan and I. S. Introduction to Spectral Theory for Selfadjoint Ordinary Differential Equations", "Nauka", Moscow, 1970, also in "Translations of Mathematical Monographs", Vol.

1. m C D(X, k) with fixed X,~-~. 3) provided that i~ ~ >i 4x -I . 10). 2. 3) holds, provided that lim inf (Im ~ )I~1-1 exp (2X I~l½) ~ Thus in the second quadrant O. 5) ~ may come~ in a sense, exponentially close to the negative real axis; this is linked to exponentially small behaviour of ImPel, ~(t) - -6(- O~). 4) is neglgible. 9). 4-5). 3. 8) . it is not necessary to stay away from the negative real axis. 4. 8. The case b = ~ x) = ik -1 + o(ikL-2). 6, it seems advantageous here to make Ikl tend to and 00 together.

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