Geodesic
Main properties of the Faddeev equation for $2 \times 2$ operator matrices
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2023), pp. 53-58.

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In the present paper we consider a $2 \times 2$ operator matrix $H$. We construct an analog of the well-known Faddeev equation for the eigenvectors of $H$ and study some important properties of this equation, related with the number of eigenvalues. In particular, the Birman–Schwinger principle for $H$ is proven.
Keywords: operator matrix, spectrum, Faddeev equation, operator valued function, Birman–Schwinger principle. 
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T. H. Rasulov; E. B. Dilmurodov. Main properties of the Faddeev equation for $2 \times 2$ operator matrices. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2023), pp. 53-58. http://geodesic.mathdoc.fr/item/IVM_2023_12_a3/

[1] Lakaev S.N., “Some spectral properties of the generalized Friedrichs model”, J. Soviet Math., 45:6 (1989), 1540–1563 | DOI | MR

[2] Friedrichs K.O., “Über die Spectralzerlegung eines Integral-operators”, Math. Ann., 115:1 (1938), 249–272 | DOI | MR | Zbl

[3] Friedrichs K.O., “On the perturbation of continuous spectra”, Comm. Pure Appl. Math., 1:4 (1948), 361–406 | DOI | MR | Zbl

[4] Motovilov A.K., Sandhas W., Belyaev Y.B., “Perturbation of a lattice spectral band by a nearby resonance”, J. Math. Phys., 42 (2001), 2490–2506 | DOI | MR | Zbl

[5] Abdullaev Zh.I., Ikromov I.A., Lakaev S.N., “O vlozhennykh sobstvennykh znacheniyakh i rezonansakh obobschennoi modeli Fridrikhsa”, Teoret. i matem. fiz., 103:1 (1995), 54–62 | MR

[6] Akchurin E.R., “O spektralnykh svoistvakh obobschennoi modeli Fridrikhsa”, Teoret. i matem. fiz., 163:1 (2010), 17–33 | DOI | Zbl

[7] Lakaev S.N, Latipov Sh.M., “O suschestvovanii i analitichnosti sobstvennykh znachenii dvukhkanalnoi molekulyarno-rezonansnoi modeli”, Teoret. i matem. fiz., 169:3 (2011), 341–351 | DOI | MR | Zbl

[8] Rasulov T.Kh., “Uravnenie Faddeeva i mestopolozhenie suschestvennogo spektra modelnogo operatora neskolkikh chastits”, Izv. vuzov. Matem., 2008, no. 12, 59–69 | Zbl

[9] Muminov M.I., Rasulov T.H., “The Faddeev equation and essential spectrum of a Hamiltonian in Fock space”, Methods Funct. Anal. Topol., 17:1 (2011), 47–57 | MR | Zbl

[10] Muminov M.I., “Vyrazhenie dlya chisla sobstvennykh znachenii modeli Fridrikhsa”, Matem. zametki, 82:1 (2007), 75–83 | DOI | MR | Zbl

[11] Rasulov T.Kh., Mukhitdinov R.T., “Konechnost diskretnogo spektra modelnogo operatora, assotsiirovannogo s sistemoi trekh chastits na reshetke”, Izv. vuzov. Matem., 2014, no. 1, 61–70 | Zbl

[12] Reed M., Simon B., Methods of Modern Mathematical Physics, v. IV, Analysis of Operators, Academic Press, New York, 1978 | MR | Zbl

[13] Glazman I.M., Direct Methods of the Qualitative Spectral Analysis of Singular Differential Operators, IPS Trans, Jerusalem, 1965 | MR