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Conditions for localization of the limit spectrum of a model operator associated with the Orr-Sommerfeld equation. (English. Russian original) Zbl 1264.34172

Dokl. Math. 86, No. 1, 549-552 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 445, No. 5, 506-509 (2012).
Consider the operator \(L_\varepsilon\) generated in \(L^2(-1,1)\) by the differential expression \[ l_\varepsilon y:=i\varepsilon^2 y''+qy \] and the boundary conditions \(y(-1)=y(1)=0\). The author verifies the following hypothesis. The limit spectral graph of the operator \(L_\varepsilon\) with potential \(q\) has the \(l_c\)-property (for definition see the paper) only if \(q\) can be analytically continued to some neighborhood of the interval \([-1,1]\).

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47E05 General theory of ordinary differential operators
34B24 Sturm-Liouville theory
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References:

[1] S. G. Reddy, P. J. Schmidt, and D. S. Henningson, SIAM J. Appl. Math. 53(1), 15–47 (1993). · Zbl 0778.34060 · doi:10.1137/0153002
[2] S. N. Tumanov and A. A. Shkalikov, Math. Notes 72, 519–526 (2002). · Zbl 1022.76016 · doi:10.1023/A:1020588429647
[3] A. A. Shkalikov, ”Spectral Portraits and the Resolvent Growth of a Model Problem Associated with the Orr-Sommerfeld Equation,” www.arXiv:math.FA/0306342v1 , 24.06.2003.
[4] V. I. Pokotilo and A. A. Shkalikov, Math. Notes 86, 442–446 (2009). · Zbl 1192.34032 · doi:10.1134/S0001434609090181
[5] E. B. Davies, Math. Z. 243, 719–743 (2003). · Zbl 1032.34078 · doi:10.1007/s00209-002-0464-0
[6] Kh. K. Ishkin, J. Math. Sci. (New York) 150, 2488–2499 (2008). · Zbl 1151.34339 · doi:10.1007/s10958-008-0147-4
[7] A. A. Shkalikov, Sovrem. Mat. Fundam. Napravl. Part 3, No. 3, 89–112 (2003).
[8] J. J. Duistermaat and F. A. Grünbaum, Commun. Math. Phys. 103, 177–240 (1986). · Zbl 0625.34007 · doi:10.1007/BF01206937
[9] V. A. Marchenko, Sturm-Liouville Operators and Their Applications (Naukova Dumka, Kiev, 1977) [in Russian]. · Zbl 0399.34022
[10] I. I. Privalov, Boundary Properties of Analytic Functions (GITTL, Moscow, 1950) [in Russian].
[11] Kh. K. Ishkin, Math. Notes 78, 64–75 (2005). · Zbl 1095.34054 · doi:10.1007/s11006-005-0100-5
[12] S. P. Novikov and I. A. Dynnikov, Russ. Math. Surv. 52, 1057–1116 (1997). · Zbl 0928.35107 · doi:10.1070/RM1997v052n05ABEH002105
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