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Let H 0 and H be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $${\varphi(H)-\varphi(H_0)}$$ for piecewise continuous functions $${\varphi}$$. This description involves the scattering matrix for the pair H, H 0.

Spectral Theory of Discontinuous Functions of Self-Adjoint Operators: Essential Spectrum