Abstract Given a self-adjoint operator and a relatively trace class perturbation, one can associate the singular spectral shift function – an integer-valued function on the real line which measures the flow of singular spectrum, not only at points outside of the essential spectrum, where it coincides with the classical notion of spectral flow, but at points within the essential spectrum too. The singular spectral shift function coincides with both the total resonance index and the singular μ-invariant. In this paper we give a direct proof of the equality of the total resonance index and singular μ-invariant assuming only the limiting absorption principle and no condition of trace class type – a context in which the existence of the singular spectral shift function is an open question. The proof is based on an application of the argument principle to the poles and zeros of the analytic continuation of the scattering matrix considered as a function of the coupling parameter.

中文翻译：

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共振指数和奇异μ-不变量

摘要 给定一个自伴随算子和一个相对迹类扰动，人们可以关联奇异谱位移函数——实线上的一个整数值函数，它测量奇异谱的流动，不仅在本质谱之外的点，它与谱流的经典概念相吻合，但也在基本谱内的点上。奇异谱位移函数与总共振指数和奇异 μ 不变量一致。在本文中，我们直接证明了总共振指数和奇异 μ 不变量的相等性，假设只有极限吸收原理，没有痕类类型的条件——奇异谱位移函数的存在是一个开放的背景问题。