Badanin, Andrey; Korotyaev, Evgeny L. Third-order operators with three-point conditions associated with Boussinesq’s equation. (English) Zbl 07305508 Appl. Anal. 100, No. 3, 527-560 (2021). Summary: We consider a non-self-adjoint third-order operator on the interval \([0,2]\) with real \(1\)-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator. Cited in 8 Documents MSC: 47E05 General theory of ordinary differential operators 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) Keywords:good Boussinesq equation; third-order operator; multi-point problem; spectral asymptotics; trace formula PDFBibTeX XMLCite \textit{A. Badanin} and \textit{E. L. Korotyaev}, Appl. Anal. 100, No. 3, 527--560 (2021; Zbl 07305508) Full Text: DOI arXiv References: [1] Eloe, PW; Hankerson, D.; Henderson, J., Positive solutions and conjugate points for multipoint boundary value problems, J Differ Equ, 95, 1, 20-32 (1992) · Zbl 0760.34022 [2] Loud, WS., Self-adjoint multi-point boundary value problems, Pac J Math, 24, 303-317 (1968) · Zbl 0176.05101 [3] Pokornyi, YuV., Zeros of the Green’s function for the de la Vallée-Poussin problem, Sb Math, 199, 6, 891-921 (2008) · Zbl 1163.34331 [4] Papanicolaou, VG., The periodic Euler-Bernoulli equation, Trans Am Math Soc, 355, 9, 3727-3759 (2003) · Zbl 1052.34079 [5] Papanicolaou, VG., The inverse periodic spectral theory of the Euler-Bernoulli equation, Dyn Partial Differ Equ, 2, 2, 127-148 (2005) · Zbl 1118.34005 [6] Belabbasi, Yu., Regularized traces of a multipoint problem for higher-order ordinary differential operators, (Russian) Differentsial’nye Uravneniya, 19, 6, 938-944 (1983) [7] McKean, H., Boussinesq’s equation on the circle, Com Pure Appl Math, 34, 599-691 (1981) · Zbl 0473.35070 [8] Kalantarov, VK; Ladyzhenskaja, OA., Formation of collapses in quasilinear equations of parabolic and hyperbolic types (Russian) boundary value problems of mathematical physics and related questions in the theory of functions, Zap Naucn Sem Leningrad Otdel Mat Inst Steklov (LOMI), 69, 77-102 (1977) [9] Dickson, R.; Gesztesy, F.; Unterkofler, K., A new approach to the Boussinesq hierarchy, Math Nachr, 198, 51-108 (1999) · Zbl 0978.35049 [10] Dickson, R.; Gesztesy, F.; Unterkofler, K., Algebro-geometric solutions of the Boussinesq hierarchy, Rev Math Phys, 11, 7, 823-879 (1999) · Zbl 0971.35065 [11] Badanin, A.; Korotyaev, E., Spectral asymptotics for the third order operator with periodic coefficients, J Differ Equ V, 253, 11, 3113-3146 (2012) · Zbl 1273.47074 [12] Badanin, A.; Korotyaev, E., A third order operator with periodic coefficients on the real line, St Petersburg Math J, 25, 713-734 (2014) · Zbl 1310.34116 [13] Deift, P.; Tomei, C.; Trubowitz, E., Inverse scattering and the Boussinesq equation, Comm, Pure Appl Math, 35, 567-628 (1982) · Zbl 0479.35074 [14] Korotyaev, E.Resonances of third order differential operators. (2016) preprint arXiv:1605.01842. · Zbl 07087770 [15] Wilder, ChE., Problems in the theory of ordinary linear differential equations with auxiliary conditions at more than two points, Trans Am Math Soc, 19, 2, 157-166 (1918) · JFM 46.0697.03 [16] Badanin, A.; Korotyaev, E., Trace formula for fourth order operators on unit interval, II Dyn PDE, 12, 3, 217-239 (2015) · Zbl 1326.47046 [17] Trubowitz, E., The inverse problem for periodic potentials, Commun Pure Appl Math, 30, 3, 321-337 (1977) · Zbl 0403.34022 [18] Korotyaev, E., Inverse Problem and the trace formula for the Hill Operator, II Math Zeitschrift, 231, 2, 345-368 (1999) · Zbl 0929.34016 [19] Birkhoff, GD., On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans Am Math Soc, 9, 2, 219-231 (1908) · JFM 39.0386.01 [20] Badanin, A.; Korotyaev, E., Even order periodic operator on the real line, Int Math Res Not, rnr057, 53 (2011) [21] Badanin, A.; Korotyaev, E., Sharp eigenvalue asymptotics for fourth order operators on the circle, J Math Anal Appl, 417, 804-818 (2014) · Zbl 1325.34100 [22] Fedoryuk, MV., Asymptotic analysis: linear ordinary differential equations (2012), Berlin, Heidelberg: Springer Science and Business Media, Berlin, Heidelberg This reference list is based on information provided by the publisher or from digital mathematics libraries. 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