2011/09/16 by M. M. Malamud, Malamud, M. M., Л. Л. Оридорога +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1109.3683
openalex publication_date 2011/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. Namely, we introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of \em regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2× 2 Dirac and Dirac type equations subject to irregular or even to degenerate boundary conditions. We emphasize that our results are the first results on the completeness problem for general first order systems even in the case of regular boundary conditions.