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

Abstract

Abstract. In the smooth scattering theory framework, we consider a pair of selfadjoint operators H 0 , H and discuss the spectral projections of these operators corresponding to the interval (−∞, λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H 0 and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty.

On the Difference of Spectral Projections

“…where f is a real-valued Borel function on R. It is also of interest to predict the smoothness of the mapping S → f (T + S) − f (T ) with respect to the smoothness of f . There is a vast amount of literature dedicated to these problems, see, e. g., Kreȋn, Farforovskaja, Peller, Birman, Solomyak, Pushnitski, Yafaev [4,9,16,17,24,25,[27][28][29][30], and the references therein.…”

“…Pushnitski [27][28][29][30] and Yafaev [30] have been studying the spectral properties of the operator D(λ) in connection with scattering theory. If the absolutely continuous spectrum of T contains an open interval and under some smoothness assumptions, the results of Pushnitski and Yafaev are applicable, cf.…”

Scattering matrix and functions of self-adjoint operators

“…Birman and M. G. Kreȋn (see also the previous works [10], [3], and [4]): det S. / D e For discontinuous functions ' the operator A.ı/ may fail to be compact; see Section 6 in [8] and [7]. In this case the essential spectrum and the absolutely continuous spectrum of A.ı/ can be explicitly described in terms of the spectrum of the scattering matrix; see [13], [14], and [15]. This fact is closely related to the subject of this work; it gives another relationship between the spectra of '.H / '.H 0 / and S. /.…”

Trace Class Conditions for Functions of Schrödinger Operators

“…(1) One of us (A. P.) studied the difference f (H ) − f (H 0 ) for functions f with jump discontinuities [29][30][31]. Among other things, it was shown that for the function f (λ) = 1 (−∞,a) (λ) with a > 0 the operator f (− + V ) − f (− ) is never compact, unless scattering at energy a is trivial.…”