arXiv subjects

Nurulla Azamov

Publications and source records attributed to Nurulla Azamov.

27 records · Page 2Linked to original sources

arXiv2016

Spectral flow and resonance index

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.

math.SP↗

arXiv2015

Spectral flow inside essential spectrum

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.

math.SP↗

arXiv2013

A constructive approach to stationary scattering theory

In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a Hilbert space $\mathcal H$ which satisfy the following conditions: (i) for any open bounded subset $Δ$ of $\mathbb R,$ the operators $F E_Δ^{H_0}$ and $F E_Δ^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with trivial kernel from $\mathcal H$ to another Hilbert space $\mathcal K$ and $J$ is a bounded self-adjoint operator on $\mathcal K.$ An example of a pair of operators which satisfy these conditions is the Schrödinger operator $H_0 = -Δ+ V_0$ acting on $L^2(\mathbb R^ν),$ where $V_0$ is a potential of class $K_ν$ (see B.\,Simon, {\it Schrödinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 \in L^\infty(\mathbb R^ν) \cap L^1(\mathbb R^ν).$ Among results of this paper is a new proof of existence and completeness of wave operators $W_\pm(H_1,H_0)$ and a new constructive proof of stationary formula for the scattering matrix. This approach to scattering theory is based on explicit diagonalization of a self-adjoint operator $H$ on a sheaf of Hilbert spaces $\EuScript S(H,F)$ associated with the pair $(H,F)$ and with subsequent construction and study of properties of wave matrices $w_\pm(λ; H_1,H_0)$ acting between fibers $\mathfrak h_λ(H_0,F)$ and $\mathfrak h_λ(H_1,F)$ of sheaves $\EuScript S(H_0,F)$ and $\EuScript S(H_1,F)$ respectively. The wave operators $W_\pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.

math-ph↗

arXiv2011

Resonance index and singular spectral shift function

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 $λ$ 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 $(λ+i0; H_0,V).$ When $λ+i0$ is shifted to $λ+iy$ 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 $(λ; H_{r_0},V)$ is the difference $N_+-N_-.$ Based on the theorem just described, a non-trivial example of singular spectral shift function is given.

math.SP↗

arXiv2010

Singular spectral shift and Pushnitski $μ$-invariant

In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $μ$-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.

math.SP↗

arXiv2010

Non-trivial singular spectral shift functions exist

In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$

math.SP↗

arXiv2007

Pushnitski's $μ$-invariant and Schrödinger operators with embedded eigenvalues

In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitski's $μ$-invariant into "absolutely continuous" and "singular" parts. As a corollary, the Birman-Krein formula follows.

math.SP↗

arXiv2007

Infinitesimal spectral flow and scattering matrix

In this note the notion of infinitesimal scattering matrix is introduced. It is shown that under certain assumption, the scattering operator of a pair of trace compatible operators is equal to the chronological exponential of the infinitesimal scattering matrix and that the trace of the infinitesimal scattering matrix is equal to the absolutely continuous part of the infinitesimal spectral flow. As a corollary, a variant of the Birman-Krein formula is derived. An interpretation of Pushnitski's $μ$-invariant is given.

math.SP↗

arXiv2007

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is established. Some examples of trace compatible affine spaces of operators are given.

math.FA↗