[Paper Review] Absolutely continuous and singular spectral shift functions

Nurulla Azamov|arXiv (Cornell University)|Oct 12, 2008

Spectral Theory in Mathematical Physics45 references16 citations

TL;DR

This paper develops a constructive framework for abstract scattering theory using the limiting absorption principle and framed Hilbert spaces, enabling explicit definitions of wave and scattering matrices almost everywhere on the real line. The key contribution is a new formula showing that the singular part of the spectral shift function is almost everywhere integer-valued, derived from a trace-class perturbation of self-adjoint operators and the Birman-Krein formula.

ABSTRACT

Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.

Motivation & Objective

• To develop a constructive approach to abstract scattering theory that overcomes limitations of conventional methods in handling continuous spectra.
• To define wave and scattering matrices unambiguously almost everywhere using the limiting absorption principle and a fixed Hilbert-Schmidt operator with trivial kernel and co-kernel.
• To establish a new formula for the scattering matrix in terms of the chronological exponential.
• To prove that the singular part of the spectral shift function is almost everywhere integer-valued, a result not previously established in this generality.
• To provide a new, constructive proof of the stationary formula for the scattering matrix and the Kato-Rosenblum theorem.

Proposed method

• Introduces a set $\Lambda(H_0;F) \subset \mathbb{R}$ of full Lebesgue measure via the limiting absorption principle, ensuring existence of wave and scattering matrices almost everywhere.
• Uses a framed Hilbert space structure to represent the absolutely continuous spectral subspace as a direct integral of fiber Hilbert spaces.
• Defines the wave matrix $w_{\pm}(\lambda;H_r,H_0)$ and scattering matrix $S(\lambda;H_r,H_0)$ via operator-valued holomorphic functions and spectral projections.
• Applies the Birman-Krein formula to decompose the spectral shift function into absolutely continuous and singular parts.
• Derives the infinitesimal spectral flow via the trace of $V E^{(a)}_{H_{0}+rV}(\lambda)$, leading to the definition of $\xi^{(a)}(\lambda)$.
• Uses the chronological exponential to express the scattering matrix as $\mathrm{T}\!\exp\left(\frac{1}{i}\int_0^1 A(r)\,dr\right)$, providing a new dynamical representation.

Experimental results

Research questions

• RQ1 Can wave and scattering matrices be defined unambiguously almost everywhere for trace-class perturbations of self-adjoint operators using a constructive framework?
• RQ2 Is the singular part of the spectral shift function necessarily integer-valued almost everywhere under trace-class perturbations?
• RQ3 Does the stationary formula for the scattering matrix hold in this new framework, and can it be derived constructively?
• RQ4 Can the spectral shift function be decomposed into absolutely continuous and singular parts via trace integrals over the perturbation path?
• RQ5 What is the relationship between the Pushnitski $\mu$-invariant and the singular spectral shift function in this setting?

Key findings

• The singular part of the spectral shift function $\xi^{(s)}(\lambda)$ is shown to be almost everywhere integer-valued, a novel result for general self-adjoint operators with trace-class perturbations.
• The absolutely continuous part of the spectral shift function is defined as $\xi^{(a)}(\lambda) = \frac{d}{d\lambda} \int_0^1 \operatorname{Tr}(V E^{(a)}_{H_{0}+rV}(\lambda))\,dr$, providing a new integral representation.
• The scattering matrix satisfies $\det S(\lambda;H_0+V,H_0) = e^{-2\pi i \xi^{(a)}(\lambda)}$ almost everywhere on $\mathbb{R}$, linking the determinant to the absolutely continuous spectral shift.
• The wave matrix satisfies a multiplicative property, and its existence is proven constructively via the limiting absorption principle and operator-valued holomorphic functions.
• The chronological exponential representation of the scattering matrix is derived, offering a new dynamical interpretation of the scattering process.
• The infinitesimal spectral flow is shown to be trace-class, and the spectral shift function's singular part is proven to be locally constant with integer jumps at resonance points.