Alexander K. Motovilov

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

ON A SUBSPACE PERTURBATION PROBLEM

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let A and V be bounded self-adjoint operators. Assume that the spectrum of A consists of two disjoint parts σ and Σ such that d = dist(σ, Σ) > 0. We show that the norm of the difference of the spectral projections E A (σ) and E A+V ({λ | dist(λ,σ) < d/2}) for A and A + V is less than one whenever either (i) ∥V∥ < 2/2+π d or (ii) ∥V∥ < 1/2 d and certain assumptions on the mutual disposition of the sets a and Σ are satisfied.

Extended Hilbert space approach to few‐body problems

A general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy‐dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

Some Sharp Norm Estimates in the Subspace Perturbation Problem

Abstract.We discuss the spectral subspace perturbation problem for a self-adjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tan Θ Theorem. We also extend the Davis–Kahan tan 2Θ Theorem to the case of some unbounded perturbations.

Ultracold collisions in the system of three helium atoms

The Faddeev differential equations for a system of three particles with a hard-core interaction are described. Numerical results on the binding energies of the 4He3 and 3He4He2 trimers and on ultracold collisions of 3,4He atoms with 4He2 dimers obtained with the help of those differential equations are reviewed. The results obtained for the hard-core model using the Faddeev equations are compared with analogous results obtained by alternative methods.

The 4He Trimer as an Efimov System

We review the results obtained in the last four decades which demonstrate the Efimov nature of the 4He three-atomic system.

Scattering length of the helium-atom-helium-dimer collision

We present our recent results on the scattering length of {sup 4}He-{sup 4}He{sub 2} collisions. These investigations are based on the hard-core version of the Faddeev differential equations. As compared to our previous calculations of the same quantity, a much more refined grid is employed, providing an improvement of about 10%. Our results are compared with other ab initio and model calculations.

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let

A Generalization of the tan 2Θ Theorem

A