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 $H.$Comment: 10 page

Singular spectral shift and Pushnitski μ\mu-invariant

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.Comment: 42 page

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 $\mu$-invariant is given.Comment: 12 pages, LaTeX; minor change

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 $\Delta$ of $\mathbb R,$ the operators $F E_\Delta^{H_0}$ and $F E_\Delta^{H_1}$ 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 Schr\"odinger operator $H_0 = -\Delta + V_0$ acting on $L^2(\mathbb R^\nu),$ where $V_0$ is a potential of class $K_\nu$ (see B.\,Simon, {\it Schr\"odinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 \in L^\infty(\mathbb R^\nu) \cap L^1(\mathbb R^\nu).$ 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 $w_\pm(\lambda; H_1,H_0)$ acting between fibers $\mathfrak h_\lambda(H_0,F)$ and $\mathfrak h_\lambda(H_1,F)$ 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.Comment: 35 page

Singular spectral shift is additive

In this note it is shown that for trace-class perturbations of self-adjoint operators the singular part of the spectral shift function is additive.Comment: Proof of the theorem in this paper contains a mistake. See in this regard section 8.3 of my paper [arXiv:0810.2072] "Absolutely continuous and singular spectral shift functions" in Diss. Math. vol. 480, 201

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.Comment: 10 pages, LaTeX; a gap in the proof of Proposition 2.4 has been fixe

A remark on imaginary part of resonance points

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.Comment: 6 pages, the new version includes a new theorem and its proo

Resonance index and singular mu-invariant

With the essential spectrum of a self-adjoint operator given a relatively trace class perturbation one can associate an integer-valued invariant which admits different descriptions as the singular spectral shift function, total resonance index, and singular $\mu$-invariant. In this paper we give a direct proof of the equality of the total resonance index and singular $\mu$-invariant assuming only the limiting absorption principle. 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.Comment: 10 page

A Topological Approach to Unitary Spectral Flow via Continuous Enumeration of Eigenvalues

It is a well-known result of T.\,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.Comment: 46 page

MATLAB based language for generating randomized multiple choice questions

In this work we describe a simple MATLAB based language which allows to create randomized multiple choice questions with minimal effort. This language has been successfully tested at Flinders University by the author in a number of mathematics topics including Numerical Analysis, Abstract Algebra and Partial Differential Equations. The open source code of Spike is available at: https://github.com/NurullaAzamov/Spike. Enquiries about Spike should be sent to [email protected]: The open source code of Spike is available at https://github.com/NurullaAzamov/Spike. Enquiries about Spike should be sent to [email protected]