Riesz bases consisting of root functions of 1D Dirac operators

2011/08/22 by Djakov, Plamen, Mityagin, Boris · 1 citation
#34L40 #47E05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1108.4225

Abstract

For one-dimensional Dirac operators Ly= i \beginpmatrix 1 amp; 0
0 amp; -1 \endpmatrix (dy)/(dx) + v y, v= \beginpmatrix 0 amp; P
Q amp; 0 \endpmatrix, y=\beginpmatrix y₁
y₂ \endpmatrix, 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 L² ([0,π], ℂ²). In particular, if the potential matrix v is skew-symmetric (i.e., Q =-P), or more generally if Q =t P for some real t ≠ 0, then there exists a Riesz basis that consists of root functions of the operator L.

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• Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter

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