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August  2018, 38(8): 3851-3873. doi: 10.3934/dcds.2018167

Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan

*Corresponding author

1Present address: Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden

Received  September 2017 Revised  February 2018 Published  May 2018

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the unit circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.

Citation: Setsuro Fujiié, Jens Wittsten. Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3851-3873. doi: 10.3934/dcds.2018167
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces , vol. 140 of Pure and Applied Mathematics, Elsevier/Academic Press, 2003.

[2]

N. Dencker, The pseudospectrum of systems of semiclassical operators, Analysis & PDE, 1 (2008), 323-373.  doi: 10.2140/apde.2008.1.323.

[3]

S. Dyatlov and M. Zworski, Stochastic stability of Pollicott-Ruelle resonances, Nonlinearity, 28 (2015), 3511-3533.  doi: 10.1088/0951-7715/28/10/3511.

[4]

J. Ecalle, Cinq Applications des Fonctions Résurgentes , Prépublications mathématiques d'Orsay, Département de mathématique, 1984.

[5]

M.A. Evgrafov and M.V. Fedoryuk, {Asymptotic behaviour as $λ\to∞$ of the solution of the equation $w''(z)-p (z, λ) w (z) = 0$ in the complex $z$-plane, Russian Mathematical Surveys, 21 (1966), 3-50. 

[6]

S. FujiiéC. Lasser and L. Nédélec, Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Analysis, 65 (2009), 17-58. 

[7]

S. Fujiié and T. Ramond, Matrice de scattering et résonances associées à une orbite hétérocline, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 31-82. 

[8]

S.V. Galtsev and A.I. Shafarevich, Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients, Theoretical and Mathematical Physics, 148 (2006), 1049-1066.  doi: 10.1007/s11232-006-0100-y.

[9]

C. Gérard and A. Grigis, Precise estimates of tunneling and eigenvalues near a potential barrier, Journal of Differential Equations, 72 (1988), 149-177.  doi: 10.1016/0022-0396(88)90153-2.

[10]

B. Grébert and T. Kappeler, Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system, Asymptotic Analysis, 25 (2001), 201-237. 

[11]

K. Hirota, Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator , Journal of Mathematical Physics, 58 (2017), 102108, 14pp. doi: 10.1063/1.4999668.

[12]

T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer Verlag, 1966.

[13]

S. Kmvissias, K. McLaughlin and P. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154), 154, Princeton University Press, 2003. doi: 10.1515/9781400837182.

[14]

E.L. Korotyaev and P. Kargaev, Estimates for periodic Zakharov-Shabat operators, Journal of Differential Equations, 249 (2010), 76-93.  doi: 10.1016/j.jde.2010.02.016.

[15]

T. Ramond, Semiclassical study of quantum scattering on the line, Communications in Mathematical Physics, 177 (1996), 221-254.  doi: 10.1007/BF02102437.

[16]

A.B. Shabat and V.F. Zakharov, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Journal of Experimental and Theoretical Physics, 34 (1972), 62-69. 

[17]

M. Zworski, Semiclassical Analysis, vol. 138 of Graduate Studies in Mathematics, American Mathematical Society, 2012. doi: 10. 1090/gsm/138.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces , vol. 140 of Pure and Applied Mathematics, Elsevier/Academic Press, 2003.

[2]

N. Dencker, The pseudospectrum of systems of semiclassical operators, Analysis & PDE, 1 (2008), 323-373.  doi: 10.2140/apde.2008.1.323.

[3]

S. Dyatlov and M. Zworski, Stochastic stability of Pollicott-Ruelle resonances, Nonlinearity, 28 (2015), 3511-3533.  doi: 10.1088/0951-7715/28/10/3511.

[4]

J. Ecalle, Cinq Applications des Fonctions Résurgentes , Prépublications mathématiques d'Orsay, Département de mathématique, 1984.

[5]

M.A. Evgrafov and M.V. Fedoryuk, {Asymptotic behaviour as $λ\to∞$ of the solution of the equation $w''(z)-p (z, λ) w (z) = 0$ in the complex $z$-plane, Russian Mathematical Surveys, 21 (1966), 3-50. 

[6]

S. FujiiéC. Lasser and L. Nédélec, Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptotic Analysis, 65 (2009), 17-58. 

[7]

S. Fujiié and T. Ramond, Matrice de scattering et résonances associées à une orbite hétérocline, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 31-82. 

[8]

S.V. Galtsev and A.I. Shafarevich, Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients, Theoretical and Mathematical Physics, 148 (2006), 1049-1066.  doi: 10.1007/s11232-006-0100-y.

[9]

C. Gérard and A. Grigis, Precise estimates of tunneling and eigenvalues near a potential barrier, Journal of Differential Equations, 72 (1988), 149-177.  doi: 10.1016/0022-0396(88)90153-2.

[10]

B. Grébert and T. Kappeler, Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system, Asymptotic Analysis, 25 (2001), 201-237. 

[11]

K. Hirota, Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator , Journal of Mathematical Physics, 58 (2017), 102108, 14pp. doi: 10.1063/1.4999668.

[12]

T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer Verlag, 1966.

[13]

S. Kmvissias, K. McLaughlin and P. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154), 154, Princeton University Press, 2003. doi: 10.1515/9781400837182.

[14]

E.L. Korotyaev and P. Kargaev, Estimates for periodic Zakharov-Shabat operators, Journal of Differential Equations, 249 (2010), 76-93.  doi: 10.1016/j.jde.2010.02.016.

[15]

T. Ramond, Semiclassical study of quantum scattering on the line, Communications in Mathematical Physics, 177 (1996), 221-254.  doi: 10.1007/BF02102437.

[16]

A.B. Shabat and V.F. Zakharov, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Journal of Experimental and Theoretical Physics, 34 (1972), 62-69. 

[17]

M. Zworski, Semiclassical Analysis, vol. 138 of Graduate Studies in Mathematics, American Mathematical Society, 2012. doi: 10. 1090/gsm/138.

Figure 1.  The configuration of Stokes lines for V (x) = cos x, with legends describing the size of Re z(x; 0, λ) for λ = 1 (left panel) and λ = 1 + 10−1i (right panel)
Figure 2.  The configuration of Stokes lines and the behavior of Re z(x; x1; λ) for V (x) = cos x, where λ = and x1 satisfies Re x1 > 0, cos x1 = µ (the turning point in the middle). The top panel describes the situation for µ = 1/2 and the bottom panel for µ = 1/2 + i/10. Branch cuts are located along (the curved edges of) the white regions.
Figure 3.  The location of amplitude base points relative the neighboring turning points $x_k(\mu)$ over a partial period for generic $V$ and $\lambda = i\mu\in B_\varepsilon(\lambda_0)$. Branch cuts are indicated by dashed lines.
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