arXiv subjects

Nicholas Hatzizisis

Publications and source records attributed to Nicholas Hatzizisis.

4 recordsLinked to original sources

arXiv2026

Semiclassical WKB Problem for the non-self-adjoint Dirac operator

We review some recent rigorous results on the semiclassical behavior ($ε\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/ε\}$ where both $A$ and $S$ are differentiable functions tending to constants as $x \to \pm \infty$. We have either employed the so-called exact WKB method, or the older WKB theory of Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables us to obtain the solution of the NLS equation via inverse scattering theory.

math.SP↗

arXiv2022

Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.

math-ph↗

arXiv2021

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

In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with a fairly smooth but not necessarily analytic potential decaying at infinity. In particular, using ideas and methods going back to Langer and Olver, we provide the complete rigorous uniform semiclassical analysis of the scattering coefficients away (slightly) from zero, and the rigorous proof of the uniform Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis is motivated by the potential applications to 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. This paper complements and extends a previous work of Fujiié and the second author, which considered a more restricted problem for a strictly analytic potential.

math-ph↗

arXiv2021

Semiclassical WKB Problem for the Non-Self-Adjoint Dirac Operator with a Multi-Humped Decaying Potential

In this paper we continue the study (initiated in arXiv:2003.13584) of the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, fairly smooth but not necessarily analytic potential decaying at infinity; in this paper we allow this potential to have several local maxima and minima. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.

math-ph↗