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Identifier

000451340

Title

Semiclassical WKB Problems for Non-self-adjoint Dirac Operators with Decaying Potentials

Author

Χατζητζήσης, Νικόλαος

Thesis advisor

Καμβύσης, Σπυρίδων

Abstract

The semiclassical analysis of the nonlinear Schr¨odinger (NLS) initial value problem in [KMM03], undertook the asymptotic ( ↓ 0) analysis of the inverse scattering problem, to which Zakharov and Shabat (see [SZ72]) have reduced the solution of the equation. There, the authors worked on the formulation of that problem as a Riemann-Hilbert factorisation problem and applied an extended version of ideas and calculations that go back to the influential work of Deift, Venakides and Zhou (confer to [DZ93] and [DVZ97]). But no careful semiclassical analysis of the direct scattering problem had been undertaken until recently (see [FK20]). Instead, an ad hoc approximation of the eigenvalues of the associated Dirac (or Zakharov-Shabat) operator by their Bohr-Sommerfeld approximants was used as a starting point and the reflection coefficient was set identically to 0, in the same formal spirit. In [FK20] the rigorous semiclassical analysis of the scattering data of (1) was completed in the case where S ≡ 0 while A is real analytic, integrable, positive and symmetric (even), with only one local maximum (at which for simplicity the second derivative is non-zero). In that work, the authors used the so-called exact WKB theory which was applied to arrive at their results 1 . Now, in the same spirit, this thesis is concerned with the rigorous semiclassical analysis of the scattering data for the following three different families/cases of (non-self-adjoint) Dirac operators: • In the first case, it is assumed that S ≡ 0 and for A the analyticity assumption of [FK20] is replaced by a mild smoothness assumption (all the other assumptions placed in that work, remain the same). Here, a different method (than that found in [FK20]) is employed; one that goes back to Langer and Olver (see [Olv75], [Olv97] and [HK21a]) 2 . • The second case is a generalization of the first. In this, S still remains identically zero but now A is allowed to have several (finitely many) local maxima and minima under the additional assumptions of possessing some smoothness and being positive. Once again, an Olver-like theory is employed (confer to [HK21b]). • The third and final case deals with a Dirac operator where A(x) = S(x) = sech(2x), x ∈ R. In this (special) case a complex variant of Olver’s method (see [Olv78]) is used that allows one to arrive at the desired results (see [FHK22]).

Language

English

Issue date

2022-07-22

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Doctoral theses

Type of Work--Doctoral theses

Permanent Link

https://elocus.lib.uoc.gr//dlib/9/f/0/metadata-dlib-1664361373-58939-24093.tkl

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