Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1604.07006

Spectral flow and resonance index

Nurulla Azamov
View original source ↗Read source document ↗Export metadata ↓

Abstract

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~$λ$ 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 $(H_0+sV-λ)^{-1}V$ corresponding to an eigenvalue $(s-r_λ)^{-1},$ where $H_0+r_λV$ 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.

Explore related subjects

math.SPmath-phmath.FAmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nurulla Azamov. 2016-07-28. Spectral flow and resonance index. https://arxiv.org/abs/1604.07006

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes

This article is concerned with mixed Steklov--Robin eigenvalues on bounded domains in $\mathbb{R}^{n}, n \geq 2$, with Lipschitz boundary. Specifically, we consider domains with symmetry of order $4$ containing a spherical hole. We obtain isoperimetric inequalities for the $k$-th Steklov-Robin eigenvalues for each $k \in \{2, 3, \dots, n+1\}$. We provide examples to emphasize the fact that the symmetry assumptions, on the family of domains considered, are crucial.

math.SP↗
arXiv2026

Steklov isospectrality of conformal metrics

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n\geq 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calderón problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide. Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schrödinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.

math.SP↗
arXiv2026

Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schrödinger Operators

We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.

math.SP↗