On the spectral resolution of a differential operator-IV

Abstract

In the present paper we investigate the asymptotic formulae involving ~(x,y,l+n) and ~(x,y,n) as a tends to  infinity, where~(x,y,1) stands for DP(H(x,y, 2) - fl(x,y,A)), H(.),X(.) being the resolution matrices associated with  two different second-order dinerential systems with the same boundary conditions at two arbitrary points o and  b. Replacing fl(x,y, A) by the resolution matrix HF(x, y.2.) of the Fourier system and then making r -y we derive  some special asymptotic formulae. A modified form of a Tauberian theorem due to Wiener plays a key role in the investigation that follows.