Geodesic
Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end
Izvestiya. Mathematics, Tome 5 (1971) no. 1, pp. 161-191 
I. S. Kats. Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end. Izvestiya. Mathematics, Tome 5 (1971) no. 1, pp. 161-191. http://geodesic.mathdoc.fr/item/IM2_1971_5_1_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the spectral function $\tau(\lambda)$ of the generalized second order boundary problem \begin{gather*} -\frac d{dM(x)}\biggl[y'_-(x)-\int_{-0}^{x-0}y(s)\,dQ(s)\biggr]-\lambda y(x)=0\qquad(0\leq x),\\ y'_-(0)=m,\qquad y(0)=n, \end{gather*} and for the function $\eta(\lambda)$, which may belong to an extremely large class of positive functions that are nonincreasing on $[1,+\infty)$, the problem of characterizing the growth of the function $\tau(\lambda)$ as $\lambda\uparrow+\infty$ and of the convergence of the integral $\int^{+\infty}\eta(\lambda)\,d\tau(\lambda)$ is connected with the behavior as $x\downarrow0$ of the function $M(x)$.

[1] Kats I. S., “O suschestvovanii spektralnykh funktsii nekotorykh singulyarnykh differentsialnykh sistem vtorogo poryadka”, Dokl. AN SSSR, 106:1 (1956), 15–18 | Zbl

[2] Kats I. S., “O povedenii spektralnykh funktsii differentsialnykh sistem vtorogo poryadka”, Dokl. AN SSSR, 106:2 (1956), 183–186

[3] Kats I. S., “O roste spektralnykh funktsii differentsialnykh sistem vtorogo poryadka”, Izv. AN SSSR. Ser. matem., 23 (1959), 257–274 | Zbl

[4] Kats I. S., “Suschestvovanie spektralnykh funktsii obobschennykh differentsialnykh sistem vtorogo poryadka s granichnymi usloviyami v singulyarnom kontse”, Matem. sb., 68(110) (1965), 174–227 | Zbl

[5] Kats I. S., “O roste spektralnykh funktsii obobschennykh granichnykh zadach vtorogo poryadka s granichnym usloviem v regulyarnom kontse”, Dokl. AN SSSR, 181:3 (1968), 534–537 | Zbl

[6] Kats I. S., “O soglasovanii koeffitsientov obobschennogo lineinogo differentsialnogo uravneniya vtorogo poryadka”, Matem. sb., 79(121) (1969), 368–380 | Zbl

[7] Kats I. S., Krein M. G., “$R$-funktsii–analiticheskie funktsii, otobrazhayuschie verkhnyuyu poluploskost v sebya”, dopolnenie 1 k knige: F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968, 629–647 | MR

[8] Kats I. S., Krein M. G., “O spektralnykh funktsiyakh struny”, dopolnenie 2 k knige: F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968, 648–737 | MR

[9] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956