半轴上复Airy算子的谱性质
Spectral properties of complex Airy operator on the semi-axis
摘要：
本文证明了Schroedinger算子$L=-d^2/dx^2+p(x)$的根函数在具有复值位势$p(x)$的半轴$\mathbb R_+$上的完备性定理。假定潜在的$p = q \pm ir$使得实函数$q$和$r$服从条件$$ q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^α, \quad α>0, $$，其中常数$c, \ c_0\in \mathbb R$、$c_1>0$和$\arg(\pm i+c) < 2απ/(2+α)$。对于Airy算子$L_c=-d^2/dx^2+cx$，$c=const$的情形，这个定理意味着这个算子的本征函数系的完备性，如果$|\arg c|<2π/3$ .使用另一种技术的基础上的Airy函数的渐近行为，我们证明了算子$L_c$的完整性定理仍然有效，提供$|\arg c|<5π/6$。
英文摘要：
We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2/dx^2+p(x)$ on the semi-axis $\mathbb R_+$ with a complex--valued potential $p(x)$. It is assumed that the potential $p = q \pm ir$ is such that the real functions $q$ and $r$ are subject the conditions $$ q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^α, \quad α>0, $$ where the constants $c, \ c_0\in \mathbb R$, $c_1>0$ and $\arg(\pm i+c) < 2απ/(2+α)$. For the case of the Airy operator $L_c=-d^2/dx^2+cx$, $c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $|\arg c|<2π/3$. Using another technique based on the asymptotic behavior of the Airy functions we prove that the completeness theorem for the operator $L_c$ remains valid, provided that $|\arg c|<5π/6$.
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