We consider the resonances of a quantum graph Ᏻ 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 Ᏻ in a disc of a large radius. We call Ᏻ a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of Ᏻ. 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.

We discuss the well known "Fredholm index=spectral flow" theorem and show that it can be interpreted as a limit case of an identity involving two spectral shift functions.

We consider the spectrum of a two-dimensional Pauli operator with a compactly supported electric potential and a variable magnetic field with a positive mean value. The rate of accumulation of eigenvalues to zero is described in terms of the logarithmic capacity of the support of the electric potential. A connection between these eigenvalues and orthogonal polynomials in complex domains is established.

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. The proof is based on the analysis of a certain new unitary invariant for a pair of self-adjoint operators. Academic Press

Abstract. The three-dimensional Schrödinger operator H with constant magnetic field of strength b > 0 is considered under the assumption that the electric potential V ∈ L 1 (R 3 ) admits certain power-like estimates at infinity. The asymptotic behavior as b → ∞ of the spectral shift function ξ(E; H, H 0 ) is studied for the pair of operators (H, H 0 ) at the energies E = Eb + λ, E > 0 and λ ∈ R being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick E/2 ∈ Z + and λ ∈ R; then the main term is always of order √ b, and is independent of λ. In the second asymptotic regime, called asymptotics near a Landau level, we choose E = 2q 0 , q 0 ∈ Z + , and λ = 0; in this case the leading term of the SSF could be of order b orThe main object of investigation in the present paper is the spectral shift function (SSF) for the three-dimensional Schrödinger operator with constant magnetic field, perturbed by an electric potential that decays sufficiently fast at infinity. We recall the abstract setting in which the SSF for a pair of selfadjoint operators occurs. First, let T 0 and T be two selfadjoint operators acting in the same Hilbert space, and let T −T 0 ∈ S 1 , where S 1 denotes the trace class. Then there exists a unique function ξ(·; T , T 0 ) ∈ L 1 (R) such that the Lifshits-Kreȋn trace formula