On localization conditions for spectrum of model operator for Orr–Sommerfeld equation

Abstract

For a model operator $L(\varepsilon)$ related with Orr-Sommerfeld equation, we study the necessity of known Shkalikov conditions sufficient for a localization of the spectrum at a graph of Y-shape. We consider two types of the potentials, for which an unbounded part $\Gamma_\infty$ of the limiting spectral graph (LSG) is constructed in an explicit form. The first of them is a piece-wise potential with countably many jumps. We show that if the discontinuity points of this potential converge rather fast to one of the end-points of the interval $(0,1)$, then $\Gamma_\infty$ consists in countably many rays. The second potential is glued from two holomorphic functions. We show that $\Gamma_\infty$ consists in two curves if the derivative at the gluing point has a jump and Langer conditions are satisfied in the domain enveloped by the Stokes lines ensuring the possibility of constructing WKB-expansions. If the gluing is infinitely differentiable, WKB-estimates are insufficient to clarify the spectral picture. Because of this we consider an inverse problem: given some spectral data, clarify analytic properties of the potential in the vicinity of the interval $(0,1)$. In order to understand the nature of spectral data, we first solve a direct problem extended to a complex $\varepsilon$-plane. It turns out that if we assume the holomorphy of the potential in the vicinity of the segment $[0,1]$, then for small $\varepsilon$ in the sector $\mathcal{E}$ of opening $\pi/2$, the part of the spectrum $L(\varepsilon)$ outside some circle satisfies quantizaion conditions of Bohr-Sommerfeld type. In the concluding part of the work we solve the inverse problem. As spectral data, quantization conditions obtained in the direct problem and taken in a slightly weaker form serve. We prove that if the potential is a monotone continuously differentiable function and the mentioned conditions are satisfied, then the potential admits an analytic continuation into some neighbourhood of the interval $(0,1)$. This proves the necessity of Shkalikov conditions at least in a local sense.