Spectral properties of complex Airy operator on the semi-axis

Abstract

We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2/d{x}^2+p(x)$ 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 $$q(x)⩾cr(x),r(x)⩾c_0+c_1{x}^{\alpha },\alpha >0,$$ where the constants $c,c_0\in \mathbb{R}$, $c_1>0$ and $\arg (\pm i+c)<2\alpha \pi /(2+\alpha )$. For the case of the Airy operator $L_c=-d^2/d{x}^2+cx$, $c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $|\arg c|<2\pi /3$. 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 $|\arg c|<5\pi /6$.

16 pages, in Russian

References in corpus (4)

• Root system of singular perturbations of the harmonic oscillator type operators
• Differential operators admitting various rates of spectral projection growth
• Spectral projections of the complex cubic oscillator
• Local form-subordination condition and Riesz basisness of root systems