The complex Airy operator with a semi-permeable barrier

TL;DR

This paper extends the complex Airy operator with a semi-permeable barrier, rigorously analyzing its spectral properties, eigenfunctions, and resolvent, providing explicit formulas and decay estimates for the associated semi-group.

Contribution

It introduces a new extension of the complex Airy operator with transmission conditions, offering a detailed spectral analysis and explicit resolvent formulas.

Findings

Spectrum is discrete

Eigenfunctions form a dense set in L^2

Explicit resolvent kernel formulas derived

Abstract

We consider a suitable extension of the complex Airy operator, $-d^2/dx^2 + ix$, on the real line with a transmission boundary condition at the origin. We provide a rigorous definition of this operator and study its spectral properties. In particular, we show that the spectrum is discrete, the space generated by the generalized eigenfunctions is dense in $L^2$ (completeness), and we analyze the decay of the associated semi-group. We also present explicit formulas for the integral kernel of the resolvent in terms of Airy functions, investigate its poles, and derive the resolvent estimates.