Duchêne, Vincent; Vukićević, Iva; Weinstein, Michael I. Homogenized description of defect modes in periodic structures with localized defects. (English) Zbl 1325.34099 Commun. Math. Sci. 13, No. 3, 777-823 (2015). The authors study bifurcation phenomena for Schrödinger operators \[ H_{Q+\lambda V} \equiv -\partial_x^2 +Q(x) + \lambda V(x), \lambda > 0, \] where \(\lambda\) is a small parameter, \(Q\) is a continuous, \(1\)-periodic function on \(\mathbb{R}\), and \(V\) is spatially localized in the sense that \(\int_{\mathbb{R}} (1+|x|)|V(x)| \;dx < \infty\). The occurrence of negative eigenvalues \(E^\lambda\) for \( H_{Q+\lambda V} \psi^\lambda = E^\lambda \psi^\lambda\) located in the spectral gap of \(H_Q\), and estimates on the eigenpair \((E^\lambda \psi^\lambda)\) are established. Reviewer: Florin Catrina (New York) Cited in 7 Documents MSC: 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators Keywords:Schrödinger operator; defect modes; effective operator; Floquet-Bloch states PDFBibTeX XMLCite \textit{V. Duchêne} et al., Commun. Math. Sci. 13, No. 3, 777--823 (2015; Zbl 1325.34099) Full Text: DOI arXiv