Literature cited
T. Carleman, Propriétés asymptotiques des fonctions fondamentals des membranes vibrates, Attende Scand. Math. Kongressen i Stockholm, pp. 34–44 (1934).
A. Pleijel, Propriétés asymptotiques des fonctions et valeurs propres de certains problemes de vibrations, Arkiv mat., astron., fyz., 27-A, No. 13, 1–100 (1940).
B. M. Levitan, On the asymptotic behavior of a spectral function and on the eigenfunction expansion of a self-adjoint second-order differential operator, Izv. AN SSSR, Ser. Matem., Vol. 19, 33–58 (1955).
B. M. Levitan; On the asymptotic behavior of a spectral function and eigenfunction expansion of an equation [in Russian], Trudy Moscow Math. Society, Vol. 4 (1955).
V. A. Marchenko, Tauberian type theorems in the spectrum analysis of differential operators, Izv. AN SSSR, Ser. Matem., Vol. 19, 381–422 (1955).
A. Haar, Zur Theorie der orthogonalen funktionalen Systeme, Math. Ann. Vol. 69, 331–339 (1910).
M. Stone, Certain integrals analogous to Fourier integrals, Math. Z., Vol. 28, 654–697 (1928).
B. M. Levitan, On a special Tauberian theorem, Izv. AN SSSR, Ser. Matem., Vol. 17, 269–284 (1953).
L. Garding, On the asymptotic properties of the spectral function belonging to a self-adjoint semibounded extension of elliptic differential operator, Kgl. fysiogr. Sallskap. Lund forhandl., Vol. 24, No. 21 (1954).
G. Bergendal, Convergence and summability of eigenfunction expansions connected with elliptic differential operators, Thesis, Lund (1959).
Wet and Mandl, On the asymptotic distribution on eigenvalues, Proc. Roy. Soc. (A), Vol. 200, 572–580 (1950).
E. Titchmarsh, On eigenfunction expansions, III, Proc. London Math. Soc.,3 (3), No. 10, 153–169 (1953).
D. Ray, On spectra of second order differential operator, Trans. Amer. Math. Soc., Vol. 77, 299–321 (1954).
I. M. Gel'fand and B. M. Levitan, On a simple identity for the eigenvalues of a second order differential operator, Doklady Akad. Nauk SSSR, Vol. 88, 593–596 (1953).
M. G. Gasymov, On the sum of differences of eigenvalues of two self-adjoint operators, Doklady Akad. Nauk SSSR, Vol. 150, No. 6 (1963).
B. I. Korenblyum, General Tauberian theorem forratios, Doklady Akad. Nauk SSSR, Vol. 88, 745–748 (1953).
V. B. Lidskii, Non-self-adjoint operator of Sturm-Liouville type with a discrete spectrum [in Russian), Trudy Moscow Math. Soc., Vol. 9 (1960).
T. Carleman, Ueber die asymptotische Verteilung der Eigenverte partieller Differentialgleichungen, Berichte der Sachsisch. Acad. Wiss. Leipzig, Math.-Phys. KL., 88.
A. G. Kostyuchenko, Estimate of the resolvent of singular elliptical operators, Doklady Akad. Nauk SSSR, Vol. 132, No. 1 (1960).
A. G. Kostyuchenko, Asymptotic distribution of eigenvalues of elliptic operators, Doklady Akad. Nauk SSSR, Vol. 158, No. 1, 42–45 (1964).
A. G. Kostyuchenko, Asymptotic of the spectral function for a singular differential operator of 2m-th order, Doklady Akad. Nauk SSSR, Vol. 168, No. 2 (1966).
A. G. Kostyuchneko, Eigenvalue distribution for singular differential operators, Doklady Akad. Nauk SSSR, Vol. 168, No. 1 (1966).
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Translated from Matematicheskie Zametki Vol. 1, No. 3, pp. 365–678, March, 1967.
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Kostyuchenko, A.G. Some spectral properties of differential operators. Mathematical Notes of the Academy of Sciences of the USSR 1, 242–250 (1967). https://doi.org/10.1007/BF01098892
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DOI: https://doi.org/10.1007/BF01098892