### Perturbation of a Sturm-Liouville Operator by a Finite Function

Richard C. Gilbert
1965 Proceedings of the American Mathematical Society
If Fi is a self-adjoint operator and F is a bounded self-adjoint operator in a Hilbert space and if F2=Fi-|-F, then Theorem 1 of [2] states that Here Ri(z) is the resolvent of F¿, || ||2 is the Schmidt norm, and 5 stands for trace. From (1) various trace formulas for differential operators may be obtained. In [2] the condition (2) was verified for the situation in which Fi is defined in L2 [0, oe ) by the ordinary differential operator L= -D2 and the boundary condition w(0) = 0, and V is the
more » ... 0, and V is the operator of multiplication by p(x), where p is real, continuous, bounded, and absolutely integrable on [0, »). Recently M. G. Gasymov [l] derived trace formulas for the case that L= -D2-\-q, where q(x)-»oo as x-»co, and p(x) is a finite1 function. Gasymov's article suggests that condition (2) is valid for the case considered by him. It is the purpose of this article to show that if p is a finite function and if F is bounded below, then (2) holds in fact, whatever the behavior of q at infinity is. The method employed is similar to that used by B. M. Levitan [3] for the study of the spectral function of L. Theorem. Let qbe a real-valued continuous function on [0, oo). Let T be a self-adjoint operator in L2[0, oo) defined by L = -D2-\-q and the boundary condition u'(0) =0. (If L is in the limit circle case at infinity, a boundary condition at infinity is also included.) Let p(x) be a realvalued continuous function on [0, co) which vanishes for x>A. If T is bounded below, \\ \ V\ ll2R(ir)\\2 = 0(r~3li) as t-»co , where Vis the operator of multiplication by p, the norm is the Schmidt norm and R(z) is the resolvent of T.