The Krein differential system and integral operators of random matrix theory
Algebra i analiz, Tome 22 (2010) no. 5, pp. 186-199.

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Earlier, the Krein differential system has been studied under certain regularity conditions. In this paper, some cases are treated where these conditions are not fulfilled. Examples related to the random matrix theory are studied.
Keywords: spectral function, scattering function, random matrix theory, triangular factorization.
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L. Sakhnovich. The Krein differential system and integral operators of random matrix theory. Algebra i analiz, Tome 22 (2010) no. 5, pp. 186-199. http://geodesic.mathdoc.fr/item/AA_2010_22_5_a7/

[1] Bellman R., Stability theory of differential equations, McGraw-Hill Book Co., Inc., New York etc., 1953 | MR | Zbl

[2] Chadan K., Sabatier P. C., Inverse problems in quantum scattering theory, Springer-Verlag, New York–Berlin, 1977 | MR | Zbl

[3] Coddington E. A., Levinson N., Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York etc., 1955 | MR | Zbl

[4] Deift P., Its A., Zhou X., “A Riemann–Hilbert approach to asymptotic problems arising in the theory of random matrix models and also in the theory of integrable statistical mechanics”, Ann. of Math. (2), 146 (1997), 149–235 | DOI | MR | Zbl

[5] Denisov S., “To the spectral theory of Kreǐn systems”, Integral Equations Operator Theory, 42 (2002), 166–173 | DOI | MR | Zbl

[6] Gokhberg I., Krein M., Teoriya volterrovykh operatorov v gilbertovom prostranstve i ee prilozheniya, Nauka, M., 1967 | MR

[7] Krein M., “Kontinualnye analogi predlozhenii o mnogochlenakh, ortogonalnykh na edinichnoi okruzhnosti”, Dokl. AN SSSR, 105:4 (1955), 637–640 | MR

[8] Krein M., “K teorii akselerant i $S$-matrits kanonicheskikh differentsialnykh sistem”, Dokl. AN SSSR, 111:6 (1956), 1167–1170 | MR

[9] Levin B. R., Teoreticheskie osnovy statisticheskoi radiotekhniki, Sov. radio, M., 1968 | Zbl

[10] Sakhnovich L. A., “Operatory, podobnye unitarnym, s absolyutno nepreryvnym spektrom”, Funkts. anal. i ego pril., 2:1 (1968), 51–63 | MR | Zbl

[11] Sakhnovich L. A., “Faktorizatsiya operatorov v $L^2(a,b)$”, Funkts. anal. i ego pril., 13:3 (1979), 40–45 | MR | Zbl

[12] Sakhnovich L. A., “Uravneniya s raznostnym yadrom na konechnom otrezke”, Uspekhi mat. nauk, 35:4 (1980), 69–129 | MR | Zbl

[13] Sakhnovich L. A., “Zadachi faktorizatsii i operatornye tozhdestva”, Uspekhi mat. nauk, 41:1 (1986), 3–55 | MR | Zbl

[14] Sakhnovich L. A., “O svoistvakh diskretnogo i nepreryvnogo spektrov radialnogo uravneniya Diraka”, Teor. i mat. fiz., 108:1 (1996), 36–49 | MR | Zbl

[15] Sakhnovich L. A., Integral equations with difference kernels on finite intervals, Oper. Theory Adv. Appl., 84, Birkhäuser Verlag, Basel, 1996 | MR | Zbl

[16] Sakhnovich L. A., “On reducing the canonical system to two dual differential systems”, J. Math. Anal. Appl., 255 (2001), 499–509 | DOI | MR | Zbl

[17] Sakhnovich L. A., Spectral theory of canonical differential systems. Method of operator identities, Oper. Theory Adv. Appl., 107, Birkhäuser Verlag, Basel, 1999 | MR | Zbl

[18] Sakhnovich L. A., “On Krein's differential system and its generalization”, Integral Equations Operator Theory, 55 (2006), 561–572 | DOI | MR | Zbl

[19] Sakhnovich L. A., “Integrable operators and canonical differential systems”, Math. Nachr., 280:1–2 (2007), 205–220 | DOI | MR | Zbl

[20] Tracy C. A., Widom H., “Introduction to random matrices”, Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992), Lecture Notes in Phys., 424, Springer, Berlin, 1993, 103–130 | MR | Zbl