Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

Abstract

In this paper we examine the semiclassical behaviour 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 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.

Complex Eigenvalue Splitting for the Dirac Operator

“…see Remark 3.5. We also obtain the following refinement of [9,Theorem 2.2] showing that for single-lobe potentials, the semiclassical eigenvalues are confined to the imaginary axis:…”

“…then it is known that the continuous spectrum of P(h) consists of the entire real axis, and that away from the origin there are no real eigenvalues. For potentials of Klaus-Shaw type satisfying (ii) , a precise description of the reflection coefficients as well as the eigenvalues close to zero has recently been obtained by Fujiié and Kamvissis [9]. Finally, it is not necessary to consider eigenvalues away from R i[−V 0 , V 0 ] since the spectrum of P(h) accumulates on this set in the limit as h → 0.…”

“…However, it is a priori not clear how such an h-dependent approximation of initial data affects the behavior of ψ as h → 0, or if it is even justified at all. For this a rigorous semiclassical description of the spectrum of the corresponding Dirac operator P(h) is required, which has so far only been provided in a few cases such as for periodic potentials by Fujiié and Wittsten [12], and for bell-shaped, even potentials by Fujiié and Kamvissis [9]. Both of the mentioned articles employ the exact WKB method which we describe in Sect.…”

Semiclassical WKB problem for the non-self-adjoint Dirac operator with a decaying potential

“…In both works we assumed the possibility of an analytic extension of a function ρ a priori defined on an imaginary interval, that gives the density of eigenvalues of the Dirac operator (accumulating on a compact interval on the imaginary axis). Eventually (see [7]) it was realized that the analyticity assumption could be discarded by use of a simple auxiliary scalar Riemann-Hilbert problem.…”

“…The rigorous analysis of this direct scattering problem was initiated in [7] (in the case of real analytic data) and more generally in [8] for data which is only required to be somewhat smooth. The rigorous analysis of the inverse scattering problem was initiated much earlier in [10] by use of an ansatz which was justified later in [11].…”