2021/09/22 by Nurula Azamov, Azamov, Nurula
Mathematics · #Spectral Theory in Mathematical Physics #Advanced Mathematical Physics Problems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2109.10545
Let~H0 and~V be self-adjoint operators such that~V admits a factorisation V = F^*JF with bounded self-adjoint J and |H0|1/2-compact~F. Flow of singular spectrum of the path of self-adjoint operators H0 + rV, r ∈ \mathbb R, -- also called spectral flow, through a point λ outside the essential spectrum of~H0 is well studied, and appears in such diverse areas as differential geometry and condensed matter physics. Inside the essential spectrum the spectral flow through λ for such a path is well-defined if the norm limit limy → 0+ F (H0 + r V - λ- iy)-1 F^* exists for at least one value of the coupling variable r ∈ \mathbb R. This raises the question: given a self-adjoint operator~H0 and |H0|1/2-compact operator F, for which real numbers λ there exists a bounded self-adjoint operator J such that the limit above exists? Real numbers λ for which this statement is true we call semi-regular and the operator V = F^*JF we call a regular direction for~H0 at λ. In this paper we prove that λ is semi-regular for~H0 if and only if the direction F^*F is regular.