Spectral properties of quasiperiodic Kronig-Penney model
Let \(H_\theta\) be the self-adjoint operator on \(L^2(\mathbb{R})\), given by \[ H_\theta:= -{d^2\over dx^2}+ \sum_{j\in\mathbb{Z}} V_\theta(j)\delta(x-j), \] where \(\delta\) denotes the one-dimensional delta-function, and \(\{V_\theta(j)\}_{j\in\mathbb{Z}}\) is a sequence which takes only two positive values \(\lambda_1\neq \lambda_2\): \[ V_\theta(j):= \lambda_1\psi_A(\Phi(\alpha j)+\theta)+ \lambda_2\psi_{A^c}(\Phi(\alpha j)+ \theta),\quad j\in\mathbb{Z}. \] Here \(\chi_A\) is the characteristic function of a subset \(A\subset \mathbb{R}/\mathbb{Z}\), \(\Phi\) is the canonical projection from \(\mathbb{R}\) onto \(\mathbb{R}/\mathbb{Z}\), \(\alpha\in (0,1)\) and \(\theta\in \mathbb{R}/\mathbb{Z}\). The authors prove the absence of the point spectrum of \(H_\theta\) under certain conditions and show that in the case of J\(A= \Phi([1- \alpha, 1])\) and \(\alpha\) irrational, \(\sigma(H_\theta)\) is independent of \(\theta\). Moreover, \(\sigma(H_\theta)\) can be represented in terms of \(\sigma(h_0(\lambda))\), where \(h_\theta(\lambda)\) is a discrete analogue of \(H_\theta\): \[ (h_\theta(\lambda) u)(n):= u(n+1)+ u(n- 1)+ \lambda v_\theta(n)u(n), \qquad v_\theta(n)= \chi_A(\Phi(\alpha n)+\theta), \] and \(\lambda\in C\), where \(C\) is a certain subset of \(\mathbb{R}\).
scientific article; zbMATH DE number 1880960