In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.
M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Vol. 55, US Government Printing Office, 1948.
2.
M.Bertola and A.Tovbis, Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquée solution to Painlevé I, Communications in Pure and Applied Mathematics66(5) (2013), 678–752. doi:10.1002/cpa.21445.
3.
P.Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, AMS, 2000.
4.
N.Dencker, The pseudospectrum of systems of semiclassical operators, Analysis & PDE1(3) (2008), 323–373. doi:10.2140/apde.2008.1.323.
5.
N.Dencker, Personal communication.
6.
S.Fujiié, N.Hatzizisis and S.Kamvissis, Semiclassical WKB problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential, Journal of Differential Equations360 (2023), 90–150. doi:10.1016/j.jde.2023.02.019.
7.
S.Fujiié and S.Kamvissis, Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential, Journal of Mathematical Physics61(1) (2020), 011510. doi:10.1063/1.5099581.
8.
S.Fujiié and J.Wittsten, Quantization conditions of eigenvalues for semiclassical Zakharov–Shabat systems on the circle, AIMS38(8) (2018), 3851–3873.
9.
N.Hatzizisis and S.Kamvissis, Semiclassical WKB problem for the non-self-adjoint Dirac operator with a decaying potential, Journal of Mathematical Physics62(3) (2021), 033510. doi:10.1063/5.0014817.
10.
K.Hirota and J.Wittsten, Complex eigenvalue splitting for the Dirac operator, Communications in Mathematical Physics, 383 (2021), 1527–1558. doi:10.1007/s00220-021-04063-5.
11.
S.Kamvissis, K.D.T.R.McLaughlin and P.D.Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Annals of Mathematics, Vol. 154, Princeton University Press, Princeton, NJ, 2003.
12.
S.Kamvissis and E.A.Rakhmanov, Existence and regularity for an energy maximization problem in two dimensions, Journal of Mathematical Physics46(8) (2005), also addendum in Journal of Mathematical Physics, v. 50, n.9, 2009.
13.
M.Klaus and J.K.Shaw, On the eigenvalues of Zakharov–Shabat systems, SIAM Journal on Mathematical Analysis34(4) (2003), 759–773. doi:10.1137/S0036141002403067.
14.
P.D.Miller, Some remarks on a WKB method for the non-selfadjoint Zakharov–Shabat eigenvalue problem with analytic potentials and fast phase, Physica D, Nonlinear Phenomena152 (2001), 145–162. doi:10.1016/S0167-2789(01)00166-X.
15.
S.Novikov, S.V.Manakov, L.P.Pitaevskii and V.E.Zakharov, Theory of Solitons: The Inverse Scattering Method, Springer Science & Business Media, 1984.
16.
F.W.J.Olver, Second-order linear differential equations with two turning points, Philosophical Transactions of the Royal Society of London, Series A, Mathematical and Physical Sciences278(1279) (1975), 137–174.
17.
F.W.J.Olver, Asymptotics and Special Functions, AK Peters/CRC Press, 1997.
18.
F.W.J.Olver, D.W.Lozier, R.F.Boisvert and C.W.Clark (eds), NIST Handbook of Mathematical Functions (Hardback and CD-ROM), Cambridge University Press, 2010.
19.
D.R.Yafaev, Passage through a potential barrier and multiple wells, St. Petersburg Mathematical Journal29(2) (2018), 399–422. doi:10.1090/spmj/1499.
20.
V.Zakharov and A.Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet physics JETP34(1) (1972), 62.