• [1] William Feller, On second order differential operators, Ann. of Math. (2) 61 (1955), 90–105. MR 0068082, https://doi.org/10.2307/1969621
• [2] William Feller, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (2) 60 (1954), 417–436. MR 0065809, https://doi.org/10.2307/1969842
• [3] Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, https://doi.org/10.1007/BF01474161
• [4] Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877
• [5] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
• [6] Kunihiko Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of 𝑆-matrices, Amer. J. Math. 71 (1949), 921–945. MR 0033421, https://doi.org/10.2307/2372377
• [7] Joanne Elliott, Eigenfunction expansions associated with singular differential operators, Trans. Amer. Math. Soc. 78 (1955), 406–425. MR 0068701, https://doi.org/10.1090/S0002-9947-1955-0068701-2
• [8] Einar Hille, The abstract Cauchy problem and Cauchy’s problem for parabolic differential equations, J. Analyse Math. 3 (1954), 81–196. MR 0062934, https://doi.org/10.1007/BF02803587
• [9] W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann. vol. 113 (1936) pp. 113-160.
• [10] Norman Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. 18 (1951), 57–71. MR 0041313
• [11] Norman Levinson, Addendum to “A simplified proof of the expansion theorem for singular second order linear differential equations.”, Duke Math. J. 18 (1951), 719–722. MR 0044705
• [12] Kôsaku Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion, Nagoya Math. J. 1 (1950), 49–58. MR 0042016
• [13] Kôsaku Yosida, Correction to my paper “On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion” in Nagoya Mathematical Journal, vol. 1 (1950), 49–58, Nagoya Math. J. 6 (1953), 187–188. MR 0058074
• [14] Samuel Karlin and James McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U. S. A. 41 (1955), 387–391. MR 0071665
• [15] W. Ledermann and G. E. H. Reuter, Spectral theory for the differential equations of simple birth and death processes, Philos. Trans. Roy. Soc. London. Ser. A. 246 (1954), 321–369. MR 0060103, https://doi.org/10.1098/rsta.1954.0001
• [16] William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468–519. MR 0047886, https://doi.org/10.2307/1969644
• [17] Einar Hille, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. MR 0025077
• [18] R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, Springer, 1931.
• [19] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
• [20] A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), no. 1, 415–458 (German). MR 1512678, https://doi.org/10.1007/BF01457949
• [21] A. Khinchin, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik u. ihrer Grenzgebiete, vol. 2 (4), New York, Chelsea, 1948.

Bibliography

[1]

W. Feller, On second order differential operators, Ann. of Math. (1) vol. 61 (1955) pp. 90-105. MR 0068082 (16:824g)

[2]

-, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (3) vol. 60 (1954) pp. 417-436. MR 0065809 (16:488d)

[3]

H. Weyl, Über gewöhnlicher Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen, Math. Ann. vol. 68 (1910) pp. 220-269. MR 1511560

[4]

M. H. Stone, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, vol. 15, New York, 1932. MR 1451877 (99k:47001)

[5]

E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, Oxford University Press, 1946. MR 0019765 (8:458d)

[6]

K. Kodaira, The eigenvalue problem for ordinary differential operators of the second order and Heisenberg's theory of -matrices, Amer. J. Math. (4) vol. 71 (1949) pp. 921-945. MR 0033421 (11:438c)

[7]

J. Elliott, Eigenfunction expansions associated with certain singular differential operators, Trans. Amer. Math. Soc. (2) vol. 78 (1955) pp. 406-425. MR 0068701 (16:927a)

[8]

E. Hille, The abstract Cauchy problem and Cauchy's problem for parabolic differential equations, Journal d'Analyse Mathématique (1) vol. 3 (1953-1954) pp. 81-196. MR 0062934 (16:45a)

[9]

W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann. vol. 113 (1936) pp. 113-160.

[10]

N. Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. (1) vol. 18 (1951) pp. 57-71. MR 0041313 (12:828f)

[11]

-, Addendum to [7], Duke Math. J. vol. 18 (1951) pp. 719-722. MR 0044705 (13:463e)

[12]

K. Yosida, On Titchmarsh-Kodaira's formula concerning Weyl-Stone's eigenfunction expansion, Nagoya Math. J. (June) vol. 1 (1950) pp. 49-58. MR 0042016 (13:39a)

[13]

-, Correction to my paper [9], Nagoya Math. J. (October vol. 6 (1953) pp. 187-188. MR 0058074 (15:317a)

[14]

S. Karlin and J. McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. U.S.A. (6) vol. 41 (1955) pp. 387-391. MR 0071665 (17:166e)

[15]

W. Ledermann and G. Reuter, Spectral theory for the differential equations of simple birth and death processes, Philos. Trans. Roy. Soc. London. Ser. A. vol. 246 (1954) pp. 321-369. MR 0060103 (15:625g)

[16]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (3) vol. 55 (1952) pp. 468-519. MR 0047886 (13:948a)

[17]

E. Hille, Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, vol. 31, New York, 1948. MR 0025077 (9:594b)

[18]

R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, Springer, 1931.

[19]

D. V. Widder, The Laplace transform, Princeton University Press, 1946. MR 0005923 (3:232d)

[20]

A. N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. vol. 104 (1931) pp. 415-458. MR 1512678

[21]

A. Khinchin, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik u. ihrer Grenzgebiete, vol. 2 (4), New York, Chelsea, 1948.