Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all
points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.
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]
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.
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 s
ingular $mu$-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.
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 spec
trum. 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.
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 $Delta$ o
f $mathbb R,$ 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 Schrodinger operator $H_0 = -Delta + V_0$ acting on $L^2(mathbb R^ u),$ where $V_0$ is a potential of class $K_ u$ (see B.,Simon, {it Schrodinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 in L^infty(mathbb R^ u) cap L^1(mathbb R^ u).$ 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.
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 s
hown 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 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 $H.$
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 Pushnitskis $mu$-invariant into absolutely continuous and singular parts. As a corollary, the Birman-Krein formula follows.
