Spectral flow inside essential spectrum VI: on essentially singular points

Abstract

Let $H_0$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$ endowed with a rigging $F,$ which is a zero-kernel closed operator from $\mathcal{H}$ to another Hilbert space $\mathcal{K}$ such that the sandwiched resolvent $F(H_0-z{)}^{-1}{F}^{\ast }$ is compact. Assume that $H_0$ obeys the limiting absorption principle (LAP) in the sense that the norm limit $F(H_0-\lambda -i0{)}^{-1}{F}^{\ast }$ exists for a.e.~$\lambda .$ Numbers~$\lambda$ for which such limit exists we call $H_0$-regular. A number~$\lambda$ we call semi-regular, if the limit $F(H_0+F^{\ast }JF-\lambda -i0{)}^{-1}{F}^{\ast }$ exists for at least one bounded self-adjoint operator $J$ on $\mathcal{K};$ otherwise we call~$\lambda$ essentially singular. In this paper I discuss essentially singular points. In particular, I give different conditions which ensure that a real number~$\lambda$ is essentially singular, and discuss their relation to eigenvalues of infinite multiplicity which are known examples of essentially singular points.

10 pages