Geodesic
Mc'Kean's transformation for 3-rd order operators
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 44-54 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a non-self-adjoint third order operator with 1-periodic coefficients. Integrating the Boussinesq equation on a circle requires solving the inverse spectral problem for this operator. In 1981, McKean introduced a transformation that reduces the spectral problem for this operator to the spectral problem for the Hill operator with a potential that depends analytically on the energy. In the present paper we are studying this transformation. 
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A. V. Badanin; E. L. Korotyaev. Mc'Kean's transformation for 3-rd order operators. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 54, Tome 533 (2024), pp. 44-54. http://geodesic.mathdoc.fr/item/ZNSL_2024_533_a2/

[1] A. Badanin, E. Korotyaev, “Hill's operators with the potentials analytically dependent on energy”, J. Diff. Eq., 271 (2021), 638–664 | DOI | MR | Zbl

[2] A. V. Badanin, E. L. Korotyaev, “Obratnaya zadacha dlya $L$-operatora pary Laksa uravneniya Bussineska na okruzhnosti”, Funkts. analiz i ego prilozh., 58:1 (2024) | Zbl

[3] A. Badanin, E. Korotyaev, Inverse problem for 3-rd order operators under the 3-point Dirichlet conditions, to be published

[4] A. Badanin, E. Korotyaev, Asymptotics of the divisor for the good Boussinesq equation, arXiv: 2409.10988

[5] E. Korotyaev, “Inverse Problem and the trace formula for the Hill Operator, II”, Mathematische Zeitschrift, 231:2 (1999), 345–368 | DOI | MR | Zbl

[6] E. Korotyaev, “Estimates of periodic potentials in terms of gap lengths”, Comm. Math. Phys., 197:3 (1998), 521–526 | DOI | MR | Zbl

[7] H. McKean, “Boussinesq's equation on the circle”, Com. Pure and Appl. Math., 34 (1981), 599–691 | DOI | MR | Zbl

[8] J. Pöschel, E.Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987 | MR | Zbl