On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

2014/01/11 by Lunyov, Anton A., Malamud, Mark M.
#34L10 #35L35 #47A15 #47E05 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1401.2574

Abstract

The paper is concerned with the completeness property of root functions of general boundary value problems for n × n first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be non-weakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval. It is also shown that the system of root vectors of the general n × n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in L²([0,1]; ℂⁿ). Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.

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