Asymptotic behavior of spectral functions of the differential operators \(-y''-\varepsilon x^2y\)
The authors consider the operator \(L\) determined by the differential expression \[ \ell y(x)\equiv -y''(x)-\varepsilon x^2y(x),\quad\varepsilon\geq 0\tag{1} \] and the boundary condition \[ y(0)\cos\alpha+ y'(t)\sin\alpha= 0,\quad\alpha\in \mathbb{R}, \tag{2} \] in the space \(L_2[0,+\infty)\). Let \(\varphi(x,\lambda)\) and \(\theta(x,\lambda)\) be the solutions of the equation \[ \ell y=\lambda y\tag{3} \] with initial conditions \[ \varphi(0,\lambda)= \theta'(0,\lambda)= \sin\alpha,\quad \varphi'(0,\lambda)= -\cos\alpha,\quad \theta(0,\lambda)= \cos\alpha.\tag{4} \] The function \[ \chi(x,\lambda)= \theta(x,\lambda)+ m(\lambda) \varphi(x,\lambda),\tag{5} \] where \(m(\lambda)\) is the Weyl-Titchmarsh function of the operator \(L\), belongs to \(L_2[0,+\infty)\) for any \(\lambda\) with \(\text{Im }\lambda> 0\). For \(\varepsilon> 0\) the spectrum of \(L\) is continuous and fills the entire real axis \(\mathbb{R}\). The derivative of the spectral function \(\rho(\lambda,\varepsilon)\) of \(L\) is continuous for all \(\lambda\in\mathbb{R}\) and is related to \(m(\lambda)\) as \[ \rho'(\lambda,\varepsilon)= {-\text{Im }m(\lambda)\over \pi}.\tag{6} \] In this paper, the authors restrict to the case \(\text{cotg }\alpha>0\) and study the behavior of \(\rho'(\lambda, \varepsilon)\) on the entire axis as \(\varepsilon\to +0\).
scientific article; zbMATH DE number 1228284