On the oscillatory behavior of singular Sturm-Liouville expansions

Abstract

A singular Sturm-Liouville operator L y = − ( P y ′ ) ′ + Q y Ly\, =\, - (Py’)’\, +\, Qy , defined on an interval [ 0 , b ∗ ) [0,b^{\ast }) of regular points, but singular at b ∗ b^{\ast } , is considered. Examples are the Airy equation on [ 0 , ∞ ) [0,\infty ) and the Legendre equation on [ 0 , 1 ) [0,1) . A mode of oscillation of the successive iterates f ( t ) f(t) , ( L f ) ( t ) (Lf)(t) , ( L 2 f ) ( 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 ⩾ 0 N\, \geqslant \, 0 , each iterate ( L k f ) ( t ) ({L^k}f)(t) shall have on ( 0 , b ∗ ) (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 ∗ b^{\ast } . An extended characterization is obtained by weakening the conditions of f at 0 and b ∗ 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 ∗ b^{\ast } .