Birkhoff regularity in terms of the growth of the norm for the Green function

E. A. Shiryaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 231-239

Résumé

We consider the ordinary differential operator $L$ generated on $[0,1]$ by the differential expression $$ l(y)=(-i)^ny^{(n)}(x)+p_2(x)y^{(n-2)}+\dots+p_{n-1}(x)y'+p_n(x)y $$ and $n$ linearly independent homogeneous boundary conditions at the endpoints. We assume that the coefficients $p_k(x)$ are Lebesgue integrable complex functions. If the boundary conditions are Birkhoff regular, then the Green function $G(\lambda)$, being the kernel of the operator $(L-\lambda)^{-1}$, admits the asymptotic estimate (for sufficiently large $|\lambda|>c_0$) $$ |G(\lambda)|\leq M|\lambda|^{\frac{-n+1}{n}}, $$ where $M=M(c_0)$ is a certain constant. In the present paper, we prove the converse assertion: the fulfillment of this estimate on some rays implies the regularity of the operator $L$.