2010/12/17 by Vadim Mogilevskii, Mogilevskii, Vadim
Materials Science · Mathematics · #34B24 #47A10 #47B25 #47E05 #FOS: Mathematics #Functional Analysis (math.FA) #Magnetism in coordination complexes #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #math.FA #msc:34B24 #msc:47A10 #msc:47B25 #msc:47E05
paper · pdf · doi:10.48550/arxiv.1012.3954
arxiv created 2010/12/17 · openalex publication_date 2010/12/17 · arxiv updated 2010/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let gH be a Hilbert space and let A be a simple symmetric operator in\n gH with equal deficiency indices d:=n_\±(A)<\∞. We show that if, for\nall l in an open interval I\⊂ bR, the dimension of defect subspaces\n gN_ l(A)(= Ker (A^*- l)) coincides with d, then every self-adjoint\nextension wt A\⊃ A has no continuous spectrum in I and the point\nspectrum of wt A is nowhere dense in I. Application of this statement to\ndifferential operators makes it possible to generalize the known results by\nWeidmann to the case of an ordinary differential expression with both singular\nendpoints and arbitrary equal deficiency indices of the minimal operator.\nMoreover, we show in the paper, that an old conjecture by Hartman and Wintner\non the spectrum of a self-adjoint Sturm - Liouville operator is not valid.\n