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Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl-Titchmarsh and spectral functions of $2\times2$ first order canonical systems. Our results improve several…
Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $\varphi(H)-\varphi(H_0)$ for piecewise continuous functions $\varphi$. This…
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to…
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms…
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.
In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of…
The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl--Titchmarsh…
We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein…
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.
We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real…
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results…
We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively…
Exact analytical expressions for the cross-section correlation functions of chaotic scattering sys- tems have hitherto been derived only under special conditions. The objective of the present article is to provide expressions that are…
We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Our formula is in terms of a regularised winding number, expressed in terms of exact differential forms, and we show…
We present a linear coordinate transform to expand the solution of scattering and emission problems into a basis of forward and backward directional vector harmonics. The transform provides intuitive algebraic and geometric interpretations…
We derive a Gutzwiller-type trace formula for quantum chaotic systems that accounts for both particle spin precession and discrete geometrical symmetries. This formula generalises previous results that were obtained either for systems with…
We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far.…
We review a recently developed transfer matrix formulation of the stationary scattering in two and three dimensions where the transfer matrix is a linear operator acting in an infinite-dimensional function space. We discuss its utility in…
Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift…
The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions…