Perturbations of the Continuous Spectrum of a Certain Nonlinear Two-Dimensional Operator Sheaf

In this paper, we consider the operator sheaf $-\Delta+V+\varepsilon\mathcal{L}_\varepsilon(\lambda)+\lambda^2$ in the space $L_2(\mathbb{R}^2)$, where the real-valued potential $V$ depends only on the first variable $x_1$, $\varepsilon$ is a small positive parameter, $\lambda$ is the spectral parameter, $\mathcal{L}_\varepsilon(\lambda)$ is a localized operator bounded with respect to the Laplacian $-\Delta$, and the essential spectrum of this operator is independent of $\varepsilon$ and contains certain critical points defined as isolated eigenvalues of the operator $-\dfrac{d^2}{dx_1^2}+V(x_1)$ in $L_2(\mathbb{R})$. The basic result obtained in this paper states that for small values of $\varepsilon$, in neighborhoods of critical points mentioned, isolated eigenvalues of the sheaf considered arise. Sufficient conditions for the existence or absence of such eigenvalues are obtained. The number of arising eigenvalues is determined, and in the case where they exist, the first terms of their asymptotic expansions are found.