The scattering matrix and the differences of spectral projections

2015

Commun. Math. Phys.

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Abstract. 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 projectionsin 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 bywhere 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 | log ε| 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.

“…this fact was established in [16,18]. In [9] it is proved that the eigenvalues of Π ε (λ) accumulate to the spectral bands (1.15) in a manner similar to (1.6).…”

Section: Examplementioning

confidence: 78%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

The Spectral Density of a Difference of Spectral Projections

2015

Commun. Math. Phys.

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“…(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.…”

Section: Setting Of the Problem In This Paper We Consider Functions mentioning

confidence: 99%

Trace Class Conditions for Functions of Schrödinger Operators

Frank

2014

Commun. Math. Phys.

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Abstract:We consider the difference f (− + V )− f (− ) of functions of Schrödinger operators in L 2 (R d ) and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions f and in V belonging only to some L p space. This is motivated by applications in mathematical physics related to Lieb-Thirring inequalities. We show that in the particular case of Schrödinger operators the well-known sufficient conditions on f , based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems forOur key idea is the use of the limiting absorption principle.

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

Integr. Equ. Oper. Theory

2010

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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.