具有半透壁的复Airy算子
The complex Airy operator with a semi-permeable barrier
摘要：
我们考虑复Airy算子的一个适当的扩展，$-d^2/dx^2 + ix$，在实直线上的传输边界条件在原点。我们给出了该算子的严格定义，并研究了其谱性质。特别是，我们表明，频谱是离散的，由广义本征函数产生的空间是密集的$L^2$（完整性），我们分析了相关的半群的衰变。我们还提出了明确的公式的积分核的预解的艾里函数，调查其极点，并推导出预解估计。
英文摘要：
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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