Riesz bases consisting of root functions of 1D Dirac operators

Details Riesz bases consisting of root functions of 1D Dirac operators Riesz bases consisting of root functions of 1D Dirac operators

2011-08-22

arxiv.org/abs/1108.4225v1

Mathematics

Spectral Theory

This placement manuscript is an extended version of the part of arXiv:1007.3234 which studies the Riesz basis property

Scientific paper

For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $L^2 ([0,\pi], \mathbb{C}^2).$ In particular, if the potential matrix $v$ is skew-symmetric (i.e., $\overline{Q} =-P$), or more generally if $\overline{Q} =t P$ for some real $t \neq 0,$ then there exists a Riesz basis that consists of root functions of the operator $L.$

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