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arXiv · 1603.06992

The complex Airy operator with a semi-permeable barrier

D. S. Grebenkov·B. Helffer·R. Henry
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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.

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D. S. Grebenkov, B. Helffer, R. Henry. 2016-03-22. The complex Airy operator with a semi-permeable barrier. https://doi.org/10.1137/16m1067408

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