Alexander Pushnitski

Spectral shift function, amazing and multifaceted

Several formula representations for the I. M. Lifshits — M. G. Kreîn spectral shift function (SSF) are discussed and intercompared. It is pointed out that the equivalence of these representations is not apparent, and different properties of the SSF are revealed by different formulas. The presentation is informal and contains no proofs.

Spectral shift function of the schrödinger operator in the large coupling constant limit

We consider the spectral shift function where H 0 is a Schrödinger operator with a variable Riemannian metric and an electromagnetic field and V is a perturbation by a multiplication operator. We prove the Weyl type asymptotic formula for in the large coupling constant limit

Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

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.

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schrödinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.

Trace formulae and high energy asymptotics for Stark operator

Abstract In , we consider the unperturbed Stark operator H0 (i.e., the Schrödinger operator with a linear potential) and its perturbation H = H 0 + Vby an infinitely smooth compactly supported potential V. The large energy asymptotic expansion for the modified perturbation determinant for the pair (H 0, H) is obtained and explicit formulae for the coefficients in this expansion are given. By a standard procedure, this expansion yields trace formulae of the Buslaev–Faddeev type.

The Spectral Density of the Scattering Matrix for High Energies

We determine the density of eigenvalues of the scattering matrix of the Schrödinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential.

ON THE HIGH-ENERGY ASYMPTOTICS OF THE INTEGRATED DENSITY OF STATES

Assuming that the integrated density of states of a Schrodinger operator admits a high energy asymptotic expansion, explicit formulae for the coefficients of this expansion are given in terms of the heat invariants.

Estimates for the spectral shift function of the polyharmonic operator

The Lifshits–Krein spectral shift function is considered for the pair of operators H0=(−Δ)l, l>0 and H=H0+V in L2(Rd), d⩾1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ>0 and the integral norms of V. These estimates are in a good agreement with the ones predicted by the classical phase space volume considerations.

Operator Theoretic Methods for the Eigenvalue Counting Function in Spectral Gaps

Abstract.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 employs some elements of the theory of the spectral shift function. 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.

The Scattering Matrix and Associated Formulas in Hamiltonian Mechanics

We prove two new identities in scattering theory in Hamiltonian mechanics and discuss the analogy between these identities and their counterparts in quantum scattering theory. These identities involve the Poincaré scattering map, which is analogous to the scattering matrix. The first of our identities states that the Calabi invariant of the Poincaré scattering map can be expressed as the regularised phase space volume. This is analogous to the Birman-Krein formula. The second identity relates the Poincaré scattering map to the total time delay and is analogous to the Eisenbud-Wigner formula.