ON THE ASYMPTOTIC BEHAVIOUR OF THE TITCHMARSH- WEYL COEFFICIENTS FOR A FOURTH ORDER EQUATION

Abstract

For second order equations -(py¹)¹ + qy = λy on [0,∞) it is known that the Titchmarsh—Wey1 coefficient m(λ) for large |λ| is asymptotically equivalent to that of the Fourier equation -y” = λy. There is at present no corresponding result for fourth order equations. In this paper we construct the _ij (λ) coefficients for the fourth order equation y(4)—((x² + 4k)y')'—¼y = λy on [O, ∞) explicitly in terms of Integrals of Whittaker functions. We show that as |λ|→∞ the coefficients approach those of the corresponding Fourier equation y⁽⁴⁾ = λy.