On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential

Kh. K. Ishkin

Ufa mathematical journal, Tome 5 (2013) no. 1, pp. 36-55

Résumé

We study the spectral properties of the operator $L_\beta$ associated with the quadratic form $\mathcal{L}_\beta=\int\limits_{0}^{\infty}(|y'|^2-\beta x^{-\gamma}|y|^2)dx$ with the domain ${Q_0=\{y\in W_2^1(0,+\infty): y(0)=0\}}$, $0\gamma2$, $\beta\in \mathbf{C}$, as well as of the perturbed operator $M_\beta=L_\beta+W$. Under the assumption $(1+x^{\gamma/2})W\in L^1(0,+\infty)$ we prove the existence of finite quantum defect of the discrete spectrum that was established earlier by L. A. Sakhnovich as $\beta>0$, $\gamma=1$ and for real $W$ satisfying a more strict decaying condition at infinity. The main result of the paper is the proof of necessity (with some reservations) of the sufficient conditions for $W(x)$ obtained earlier by Kh. Kh. Murtazin under which the Weyl function of the operator $M_\beta$ possesses an analytic continuation on some angle from non-physical sheet.

Détail

Keywords: spectral instability, localization of spectrum, quantum defect, Weyl function
Mots-clés : Darboux transformation.

Bibliographie

[1] Keldysh M. V., “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh klassov nesamosopryazhennykh uravnenii”, DAN SSSR, 77:1 (1951), 11–14 | Zbl

[2] Keldysh M. V., “Ob odnoi tauberovoi teoreme”, Trudy matem. in-ta im. V. A. Steklova, 38, 1951, 77–86 | Zbl

[3] Korenblyum B. I., “Obschaya tauberova teorema dlya otnosheniya funktsii”, DAN SSSR, 88:5 (1953), 745–748 | MR

[4] Agranovich M. S., “Spektralnye svoistva zadach difraktsii”: Voitovich N. N., Katsenelenbaum V. Z., Sivov A. N., Obobschennyi metod sobstvennykh kolebanii v teorii difraktsii, M., 1977 | MR

[5] Markus A. S., Matsaev V. I., “Teoremy sravneniya spektrov lineinykh operatorov i spektralnye asimptotiki”, Tr. Moskovskogo matematicheskogo obschestva, 1982

[6] Shkalikov A. A., “O bazisnosti kornevykh vektorov vozmuschennogo samosopryazhennogo operatora”, Teoriya funktsii i differentsialnye uravneniya, Sb. statei. K 105-letiyu so dnya rozhdeniya akademika Sergeya Mikhailovicha Nikolskogo, Tr. MIAN, 269, M., 2010, 290–303 | MR | Zbl

[7] Sadovnichii V. A., Podolskii V. E., “Sledy operatorov”, UMN, 61(371):5 (2006), 89–156 | DOI | MR | Zbl

[8] Davies E. B., “Non-self-adjoint differential operators”, Bull. London Math. Soc., 34:5 (2002), 513–532 | DOI | MR | Zbl

[9] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[10] Ambarzumian V. A., “Überline Frage der Eigenwerttheorie”, Zs. f. Phys., 53 (1929), 690–695 | DOI | Zbl

[11] Ishkin Kh. K., “O kriterii lokalizatsii sobstvennykh chisel spektralno neustoichivogo operatora”, Dokl. RAN, 429:3 (2009), 301–304 | MR | Zbl

[12] Ishkin Kh. K., “Ob usloviyakh lokalizatsii predelnogo spektra modelnogo operatora, svyazannogo s uravneniem Orra–Zommerfelda”, Dokl. RAN, 445:5 (2012), 506–509 | Zbl

[13] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 4, Mir, M., 1982

[14] Naimark M. A., “Issledovaniya spektra i razlozheniya po sobstvennym funktsiyam nesamosopryazhennogo differentsialnogo operatora 2-go poryadka na poluosi”, Tr. Mosk. matem. ob-va, 3, 1954, 181 – 270 | MR | Zbl

[15] Lyantse V. E., “O differentsialnom operatore so spektralnymi osobennostyami, I”, Matem. sb., 64(106):4 (1964), 521–561

[16] Lyantse V. E., “O differentsialnom operatore so spektralnymi osobennostyami, II”, Matem. sb., 65(107):1 (1964), 47–103

[17] Pavlov B. S., “O nesamosopryazhennom operatore Shredingera na poluosi. I–III”, Problemy matematicheskoi fiziki, 1966, no. 1, 102–132 ; 1967, No 2, 102–132; 1968, No 3, 59–80 | MR | Zbl | Zbl

[18] Murtazin Kh. Kh., “O svoistvakh rezolventy differentsialnogo operatora s kompleksnymi koeffitsientami”, Mat. zametki, 31:2 (1982), 231–244 | MR | Zbl

[19] Sakhnovich L. A., “O spektre radialnogo uravneniya Shredingera v okrestnosti nulya”, Matem. sb., 67(109):2 (1965), 221–243

[20] Glazman I. M., Pryamye metody kachestvennogo spektralnogo analiza singulyarnykh differentsialnykh operatorov, Fizmatgiz, M., 1963 | MR

[21] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 2, Mir, M., 1978 | MR

[22] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 1, Mir, M., 1977 | MR

[23] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR | Zbl

[24] Murtazin Kh. Kh., Amangildin T. G., “Asimptotika spektra operatora Shturma–Liuvillya”, Matem. sb., 110(152):1 (1979), 135–149 | MR | Zbl

[25] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[26] Zommerfeld A., Stroenie atoma i spektry, v. 1, Gostekhizdat, M., 1956

[27] Ishkin Kh. K., Murtazin Kh. Kh., “O kvantovom defekte operatora Diraka s neanaliticheskim potentsialom”, TMF, 125:3 (2000), 444–452 | DOI | MR | Zbl

[28] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M.–L., 1950

[29] Ishkin Kh. K., “O spektralnoi neustoichivosti operatora Shturma–Liuvillya s kompleksnym potentsialom”, Diff. uravneniya, 45:4 (2009), 480–495 | MR | Zbl

[30] Levitan B. M., Sargsyan I. S., Operatory Shturma–Liuvillya i Diraka, Nauka, M., 1988 | MR | Zbl

[31] Yurko V. A., Vvedenie v teoriyu obratnykh spektralnykh zadach, Fizmatlit, M., 2007 | Zbl

[32] Duistermaat J. J., Grünbaum F. A., “Differential equations in the spectral parameter”, Commun. Math. Phys., 103 (1986), 177–240 | DOI | MR | Zbl

[33] Oblomkov A. A., “Bezmonodromnye operatory Shredingera s kvadratichno rastuschim potentsialom”, TMF, 121:3 (1999), 374–386 | DOI | MR | Zbl

[34] Ishkin Kh. K., “O neobkhodimykh usloviyakh lokalizatsii spektra zadachi Shturma–Liuvillya na krivoi”, Mat. Zametki, 78:1 (2005), 72–84 | DOI | MR | Zbl