Geodesic
Birkhoff regularity in terms of the growth of the norm for the Green function
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 231-239.

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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 $L$. 
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E. A. Shiryaev. Birkhoff regularity in terms of the growth of the norm for the Green function. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 231-239. http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a13/

[1] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[2] Tamarkin Ya. D., O nekotorykh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii i o razlozhenii proizvolnykh funktsii v ryady, Petrograd, 1917 | Zbl

[3] Shkalikov A. A., “Kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Tr. seminara im. I. G. Petrovskogo, no. 9, 1983, 190–229 | MR | Zbl

[4] Birkhoff G., “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Amer. Math. Soc., 9 (1908), 373–395 | DOI | MR | Zbl

[5] Minkin A. M., Regularity of dissipative operators, , 1999 http: arXiv:math.SP/9909092 | MR

[6] Minkin A. M., Resolvent's growth and Birkhoff-regularity, To be published in JMAA