Essential spectra of one-dimensional Dirac operators
The authors introduce a method for calculating the essential spectrum of one-dimensional Dirac operators associated to differential expressions of the form \[ D = W^{-1}\Big\{J\frac{d}{dt} - P \Big\},\quad W=\mathrm{diag}(w_1,w_2), \] where \(w_j(t)>0\) and \(P\) is a self-adjoint \(2\times 2\) matrix of functions. The entries of \(P\) and the \(w_j\) are assumed to be locally integrable on the interval \((a,b)\). In particular, these assumptions permit singularities at the endpoints \(a\) and \(b\). To study the essential spectrum of the operator \(L\) generated by \(D\), the authors introduce two pencils of Sturm-Liouville operators \(\tau_1(\lambda)\) and \(\tau_2(\lambda)\) and determine subsets and supersets for \(\sigma_{\mathrm{ess}}(L)\) using the pencils. For example, setting \[ \sigma_j = \big\{\lambda\in \mathbb R \mid 0\notin \sigma_{\mathrm{ess}}(\tau_j(\lambda))\big\},\quad j=1,2, \] the authors prove \[ \sigma_1\cup \sigma_2 \subset \sigma_{\mathrm{ess}}(L). \] The paper concludes with a section devoted entirely to examples. Each example illustrates how the abstract results developed in the paper can be applied to determine the essential spectrum of various explicit Dirac operators precisely.
scientific article; zbMATH DE number 6149033
scientific article