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Register a new userSpectral flow and resonance index
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Nurulla Azamov
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It has been shown recently that spectral flow admits a natural integer-valued extension to essential spectrum. This extension admits four different interpretations; two of them are singular spectral shift function and total resonance index. In this work we study resonance index outside essential spectrum. Among results of this paper are the following. 1. Total resonance index satisfies Robbin-Salamon axioms for spectral flow. 2. Direct proof of equality total resonance index = intersection number. 3. Direct proof of equality total resonance index = total Fredholm index. 4. (a) Criteria for a perturbation~$V$ to be tangent to the~resonance set at a point~$H,$ where the resonance set is the infinite-dimensional variety of self-adjoint perturbations of the initial self-adjoint operator~$H_0$ which have~$lambda$ as an eigenvalue. (b) Criteria for the order of tangency of a perturbation~$V$ to the resonance set. 5. Investigation of the root space of the compact operator $(H_0+sV-lambda)^{-1}V$ corresponding to an eigenvalue $(s-r_lambda)^{-1},$ where $H_0+r_lambda V$ is a point of the resonance set. This analysis gives a finer information about behaviour of discrete spectrum compared to spectral flow. Finally, many results of this paper are non-trivial even in finite dimensions, in which case they can be and were tested in numerical experiments.
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This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) notion introduced in this paper. Resonance index can be described as follows. For a fixed $lambda$ the resonance points $r_0$ of a path $H_r$ of self-adjoint operators are real poles of a certain meromorphic function associated with the triple $(lambda+i0; H_0,V).$ When $lambda+i0$ is shifted to $lambda+iy$ with small $y>0,$ that pole get off the real axis in the coupling constant complex plane and, in general, splits into some $N_+$ poles in the upper half-plane and some $N_-$ poles in the lower half-plane (counting multiplicities). Resonance index of the triple $(lambda; H_{r_0},V)$ is the difference $N_+-N_-.$ Based on the theorem just described, a non-trivial example of singular spectral shift function is given.
In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $mu$-invariant does not depend on the angle variable. This gives an alternative proof of integer-valuedness of the singular part of the spectral shift function. As a consequence, the Birman-Krein formula for trace class perturbations follows.
In this paper we prove for rank one perturbations that negative two times reciprocal of the imaginary part of resonance point is equal to the rate of change of the scattering phase as a function of the coupling constant, where the coupling constant is equal to the real part of the resonance point. This equality is in agreement with Breit-Wigner formula from quantum scattering theory. For general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of resonance points, non-real and real.
The spectral flow is a classical notion of functional analysis and differential geometry which was given different interpretations as Fredholm index, Witten index, and Maslov index. The classical theory treats spectral flow outside the essential spectrum. Inside essential spectrum, the spectral shift function could be considered as a proper analogue of spectral flow, but unlike the spectral flow, the spectral shift function is not an integer-valued function. In this paper it is shown that the notion of spectral flow admits a natural integer-valued extension for a.e. value of the spectral parameter inside essential spectrum too and appropriate theory is developed. The definition of spectral flow inside essential spectrum given in this paper applies to the classical spectral flow and thus gives one more new alternative definition of it.
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.
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