On the spectrum of a class of one-dimensional Schrödinger type operators with generalized potential (Q1277561)

Let \(Q(x)=p(x)\) for \(x\leq 0\) and \(q(x)\) for \(x>0\), where \(p\) is a continuous real valued function such that \(\int_{-\infty}^0| p(x)| dx<\infty\), and \(q\) is a continuous real valued periodic function of period \(1\) on the half-line. Consider the set of \(L_2(-\infty,\infty)\) functions \(D(H)=\{g: g\in W_2^2({\mathbb{R}}| {\mathbb{N}})\cap W_2^1({\mathbb{R}}), g'(n+0)-g'(n-0)=\varepsilon g(n)\}\) for \(n\in{\mathbb{N}}\). Here \(\varepsilon\) is a real number. The author characterizes the eigenvalues and the continuous spectrum of the Schrödinger operator \(Hf=-f''(x)+Q(x)f(x)\) for \(f\in D(H)\). The continuous spectrum is bounded from below. The potential \(Q(x,\varepsilon)=p(x)\) for \(x\leq 0\) and \(q(x)+\varepsilon\sum_{n=1}^\infty\delta(x-n)\) for \(x>0\) is used. Here \(\delta(x)\) is the Dirac messure.