Spectral flow inside essential spectrum IV: $F^{\ast }F$ is a regular direction

Abstract

Let~$H_0$ and~$V$ be self-adjoint operators such that~$V$ admits a factorisation $V=F^{\ast }JF$ with bounded self-adjoint $J$ and $|H_0{|}^{1/2}$-compact~$F.$ Flow of singular spectrum of the path of self-adjoint operators $H_0+rV,$ $r\in \mathbb{R},$ -- also called spectral flow, through a point $\lambda$ outside the essential spectrum of~$H_0$ is well studied, and appears in such diverse areas as differential geometry and condensed matter physics. Inside the essential spectrum the spectral flow through $\lambda$ for such a path is well-defined if the norm limit $$\underset{y\to 0^{+}}{\lim }F(H_0+rV-\lambda -iy{)}^{-1}{F}^{\ast }$$ exists for at least one value of the coupling variable $r\in \mathbb{R}$. This raises the question: given a self-adjoint operator~$H_0$ and $|H_0{|}^{1/2}$-compact operator $F,$ for which real numbers $\lambda$ there exists a bounded self-adjoint operator $J$ such that the limit above exists? Real numbers $\lambda$ for which this statement is true we call semi-regular and the operator $V=F^{\ast }JF$ we call a regular direction for~$H_0$ at $\lambda .$ In this paper we prove that $\lambda$ is semi-regular for~$H_0$ if and only if the direction $F^{\ast }F$ is regular.

4 pages

References in corpus (1)

• Absolutely continuous and singular spectral shift functions