Research output: Contribution to journal › Article

Original language | English |
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Pages (from-to) | 1153-1181 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 338 |

Issue number | 3 |

Early online date | 5 Jun 2015 |

DOIs | |

Publication status | Published - Sep 2015 |

**The Spectral Density of a Difference_PHUSHNITSKI_Accepted 4Mar2015_GREEN AAM**density3_2.pdf, 412 KB, application/pdf

17/05/2016

Accepted author manuscript

Let H_{0} and H be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if λ belongs to the absolutely continuous spectrum of H_{0} and H, then the difference of spectral projections D(λ)=1(-∞,0)(H-λ)-1(-∞,0)(H_{0}-λ)in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations D_{ε}(λ) of D(λ), given by D_{ε}(λ)=ψ_{ε}(H-λ)-ψ_{ε}(H_{0}-λ),where ψ_{ε}(x)=ψ(x/ε) and ψ(x) is a smooth real-valued function which tends to 1/2 as x → ±∞. We prove that the eigenvalues of D_{ε}(λ) concentrate to the absolutely continuous spectrum of D(λ) as ε → +0. We show that the rate of concentration is proportional to |logε| and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of ψ. The proof relies on the analysis of Hankel operators.

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