Asymptotic Estimates for the Sturm Liouville Spectrum

Harry Hochstadt

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able ^k+1 ' '^k " ° V^^l A result due to Erdfelyi .
Erdelyi derived ^ -^ some estimates for small (;(x) + ••• and for X. We use X = hk^ + en + ....
22 - Retaining only € terms we have immediately ^^ l6k Use of the boundary conditions i(0) = ^(f) = and the Fourier series 0(x) = >; n=l a cos 2mc n shows that M- = ^2k 2 so that . V2 '^2k X = ilk + — 7:— - Similarly for case 1. ) we find ) v2 '^2k X = 4k - — TT- so that 23 l\+l -^1 = l^^2kl It is evident from the previous development that Erde...lyi's result is not correct in general, hut Magnus has shown'- -' that in an average sense it is correct. If 7^

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