Operator theoretic methods for the eigenvalue counting function in spectral gaps

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 function theory. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman-Schwinger principle.Comment: Latex, 25 page

Scattering matrix and functions of self-adjoint operators

In the scattering theory framework, we consider a pair of operators $H_0$, $H$. For a continuous function $\phi$ vanishing at infinity, we set $\phi_\delta(\cdot)=\phi(\cdot/\delta)$ and study the spectrum of the difference $\phi_\delta(H-\lambda)-\phi_\delta(H_0-\lambda)$ for $\delta\to0$. We prove that if $\lambda$ is in the absolutely continuous spectrum of $H_0$ and $H$, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix $S(\lambda)$ for the pair $H_0$, $H$ and (ii) the singular values of the Hankel operator $H_\phi$ with the symbol $\phi$

The spectral shift function and the invariance principle

The new representation formula for the spectral shift function due to F.Gesztesy and K.A.Makarov is considered. This formula is extended to the case of relatively trace class perturbations.Comment: LaTeX 2e, 38 page

An integer-valued version of the Birman-Krein formula

We discuss an identity in abstract scattering theory which can be interpreted as an integer-valued version of the Birman-Krein formula.Comment: 8 pages, Latex. To appear in the M.Sh.Birman memorial issue of Functional Analysis and Its Application

The spectral density of a difference of spectral projections

Let $H_0$ and $H$ be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if $\lambda$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(\lambda)=1_{(-\infty,0)}(H-\lambda)-1_{(-\infty,0)}(H_0-\lambda)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(\lambda)$ of $D(\lambda)$, given by $$D_\varepsilon(\lambda)=\psi_\varepsilon(H-\lambda)-\psi_\varepsilon(H_0-\lambda),$$ where $\psi_\varepsilon(x)=\psi(x/\varepsilon)$ and $\psi(x)$ is a smooth real-valued function which tends to $\mp1/2$ as $x\to\pm\infty$. We prove that the eigenvalues of $D_\varepsilon(\lambda)$ concentrate to the absolutely continuous spectrum of $D(\lambda)$ as $\varepsilon\to+0$. We show that the rate of concentration is proportional to $|\log\varepsilon|$ and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of $\psi$. The proof relies on the analysis of Hankel operators.Comment: Final version; to appear in Commun. Math. Physic

Non-Weyl Resonance Asymptotics for Quantum Graphs

We consider the resonances of a quantum graph $\mathcal G$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal G$ in a disc of a large radius. We call $\mathcal G$ a \emph{Weyl graph} if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathcal G$. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure

Inverse spectral theory for a class of non-compact Hankel operators

We characterize all bounded Hankel operators $\Gamma$ such that $\Gamma^*\Gamma$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data