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Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential. (English) Zbl 1431.81059
Summary: In this paper, we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic nonlinear Schrödinger (NLS) equation in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.
©2020 American Institute of Physics

MSC:
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81R25 Spinor and twistor methods applied to problems in quantum theory
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
81U40 Inverse scattering problems in quantum theory
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[1] Kamvissis, S.; McLaughlin, K. D. T.-R.; Miller, P. D., Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (2003), Princeton University Press: Princeton University Press, Princeton, NJ · Zbl 1057.35063
[2] Kamvissis, S.; Rakhmanov, E. A., Existence and regularity for an energy maximization problem in two dimensions, J. Math. Phys., 46, 8, 083505 (2005) · Zbl 1110.81083
[3] Kamvissis, S., On the Analyticity of the Spectral Density for Semiclassical NLS, 2002-2043 (2002), Max Planck Institute Preprint
[4] Kamvissis, S., Comment on “Existence and regularity for an energy maximization problem in two dimensions” [S. Kamvissis and E. A. Rakhmanov, J. Math. Phys. 46, 083505 (2005)], J. Math. Phys., 50, 104101 (2009) · Zbl 1283.81072
[5] Lyng, G.; Miller, P. D., The N-soliton of the focusing nonlinear Schrödinger equation for N large, Commun. Pure Appl. Math., 60, 951-1026 (2007) · Zbl 1185.35259
[6] Bertola, M.; Tovbis, A., Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquée solution to Painlevé I, Commun. Pure Appl. Math., 66, 5, 678-752 (2013) · Zbl 1355.35169
[7] Brooke Benjamin, T.; Feir, J. E., The disintegration of wave trains on deep water. Part 1. Theory, J. Fluid Mech., 27, 3, 417-430 (1967) · Zbl 0144.47101
[8] Biondini, G.; Mantzavinos, D., Universal nature of the nonlinear stage of modulational instability, Phys. Rev. Lett., 116, 043902 (2016) · Zbl 1356.35214
[9] Madelung, E., Quantentheorie in hydrodynamischer form, Z. Phys., 40, 3-4, 322-326 (1927) · JFM 52.0969.06
[10] Ecalle, J., Cinq Applications des Fonctions Résurgentes (1984), Prépublication Orsay
[11] Voros, A., The return of the quartic oscillator: The complex WKB method, Ann. Inst. H. Poincaré, 29, 211-338 (1983) · Zbl 0526.34046
[12] Gérard, C.; Grigis, A., Precise estimates of tunneling and eigenvalues near a potential barrier, J. Differ. Equations, 72, 149-177 (1988) · Zbl 0668.34022
[13] Fujiié, S.; Lasser, C.; Nédélec, L., Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Anal., 65, 1-2, 17-58 (2009) · Zbl 1191.35107
[14] Klaus, M.; Shaw, J. K., On the eigenvalues of Zakharov-Shabat systems, SIAM J. Math. Anal., 34, 4, 759-773 (2003) · Zbl 1034.34097
[15] Hirota, K. and Wittsten, J., “Complex eigenvalue splitting for the Zakharov-Shabat operator,” .
[16] Klaus, M.; Shaw, J. K., Purely imaginary eigenvalues of Zakharov-Shabat systems, Phys. Rev. E, 65, 036607 (2002)
[17] Fujiié, S.; Ramond, T., Matrice de scattering et résonances asociées à une orbite hétérocline, Ann. Inst. H. Poincaré Phys. Théor., 69, 1, 31-82 (1998) · Zbl 0916.34071
[18] Klaus, M., Eigenvalue asymptotics for Zakharov-Shabat systems with long-range potentials, Oper. Matrices, 12, 1, 55-106 (2018) · Zbl 1394.34182
[19] As we explain in Refs. 3 and 26, this corresponds to different sheets of the logarithmic kernel in this integral. Different approximations are required in different sheets for best results.
[20] We could still ignore the improvement in Ref. 6 and give a different argument involving different circles in different steps of the Riemann-Hilbert sequence. We feel that the argument would become a bit more cumbersome.
[21] The function θ^0(λ) = iπλ + πA of Ref. 6 has to be replaced by the integral of the eigenvalue density in the general case.
[22] In Ref. 14, the exact estimate is stated in the abstract, but a proof is only presented for the case of A with compact support. Still, the proof presented easily generalizes for the case of non-compact support. In fact, the crucial integral in (2.15) of Ref. 14 is positive also in the non-compact support case (as is proved, for example, in Ref. 18), and this, in turn, implies the exact estimate for the number of eigenvalues.
[23] Faddeev, L.; Takhtajan, L., Hamiltonian Methods in the Theory of Solitons (1987), Springer
[24] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media, J. Exp. Theor. Phys., 34, 1, 62-69 (1972)
[25] Deift, P.; Kamvissis, S.; Kriecherbauer, T.; Zhou, X., The toda rarefaction problem, Commun. Pure Appl. Math., 49, 1, 35-83 (1996) · Zbl 0857.34025
[26] Kamvissis, S., From stationary phase to steepest descent, Contemp. Math., 458, 145-162 (2008) · Zbl 1155.37042
[27] Its, A. R., Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromic deformations of systems of linear differential equations, Sov. Math. Doklady, 24, 3, 14-18 (1982)
[28] Jimbo, M.; Miwa, T.; Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Physica D, 2, 2, 306-352 (1981) · Zbl 1194.34167
[29] Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems: Asymptotics for the MKdV equation, Ann. Math., 137, 2, 295-368 (1993) · Zbl 0771.35042
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