On the oscillatory behavior of singular Sturm-Liouville expansions

Shaw, J. K.

doi:10.1090/S0002-9947-1980-0552270-9

On the oscillatory behavior of singular Sturm-Liouville expansions
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Trans. Amer. Math. Soc. 257 (1980), 483-505 Request permission

Abstract:

A singular Sturm-Liouville operator $Ly = - (Py’)’ + Qy$, defined on an interval $[0,b^{\ast })$ of regular points, but singular at $b^{\ast }$, is considered. Examples are the Airy equation on $[0,\infty )$ and the Legendre equation on $[0,1)$. A mode of oscillation of the successive iterates $f(t)$, $(Lf)(t)$, $({L^2}f)(t), \ldots$ of a smooth function f is assumed, and the resulting influence on f is studied. The nature of the mode is that for a fixed integer $N \geqslant 0$, each iterate $({L^k}f)(t)$ shall have on $(0,b^{\ast })$ exactly N sign changes which are stable, in a certain sense, as k varies. There is quoted from the literature the main characterization of such functions f which additionally satisfy strong homogeneous endpoint conditions at 0 and $b^{\ast }$. An extended characterization is obtained by weakening the conditions of f at 0 and $b^{\ast }$. The homogeneous endpoint conditions are replaced by a summability condition on the values, or limits of values, of f at 0 and $b^{\ast }$.

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