Yu. A. Kuperin

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.

Resonances and the extension of dynamics to rigged Hilbert space

Abstract We propose a unified operator theoretic formulation of resonances and resonance states in rigged Hilbert space. Our approach shows how resonances are the basis for the extension of dynamics for unstable systems, and in a synthesis and a generalization of the work of the Brussels/Austin groups directed by Prigogine with the work of Bohm and Gadella on Gamow vectors and with the extension theory developed by the St. Petersburg group directed by Pavlov. We illustrate the approach for prototypes of resonances, namely resonances of the power spectrum, scattering resonances and resonances associated with the unstable Bloch states.

Model of resonance scattering of composite particles

Методом теории расширений нерелятивистского гамильтониана с выходом в дополнительное пространство внутренних степеней свободы построена модель бинарных реакций в системе частиц, обладающих нетривиальной внутренней структурой. Модель применяется для описания адрон-адронного рассеяния при низких и промежуточных энергиях.

One-dimensional model of three-particle resonances

G помощью теории расширений построена нетривиальная матрица рассеяния для системы трех одномерных частиц в модели граничных условий. Исследована аналитическая структура построенной ^-матрицы на энергетической поверхности.

Coulomb two-body problem with internal structure

Методами теории расширений с выходом из основного гильбертова пространства построена модель взаимодействия внешнего (кулоновского) и внутреннего (кваркового) каналов в задаче двух тел. Изучено взаимное влияние спектров соответствующих канальных гамильтонианов, приводящее, в частности, к перестройке спектров адронных атомов. Получено явное представление для ^-матрицы и изучены ее особенности на энергетической поверхности

Scattering on the hyperbolic plane in the Aharonov-Bohm gauge field

The scattering of a quantum charged particle on the hyperbolic plane in the Abelian Aharonov-Bohm field is considered. S-matrix, scattering amplitude and its flat space limit are constructed in a closed form by the Sommerfeld integral approach.

An extensions theory setting for scattering by breathing bag

A model of the scattering of a structureless pointlike particle on a spherical bag with an internal structure, imitating ‘‘quark’’ degrees of freedom, is considered. It is assumed that the bag is a dynamical object and its radius plays the role of additional dynamical variable. The energy of the collision is distributed among the quark excitations and the vibration degrees of freedom of the bag surface. In the frame of the theory of extensions the Hamiltonian of the coupled bag‐quark system interacting with the pointlike particle is constructed. The formal multichannel S matrix of the problem is obtained.

Quantum few-body problem with internal structure. I. Two-body problem

The methods of the theory of extensions to an auxiliary Hilbert space are used to construct in the two-particle sector a scattering theory for particles possessing internal structure. The analytic properties of the amplitudes of resonance scattering and Greens functions corresponding to a class of singular energy-dependent interactions are investigated.

Solvable model of spin-dependent transport through a finite array of quantum dots

The problem of spin-dependent transport of electrons through a finite array of quantum dots attached to a 1D quantum wire (spin gun) for various semiconductor materials is studied. The Breit–Fermi term for spin–spin interaction in the effective Hamiltonian of the device is shown to result in a dependence of transmission coefficient on the spin orientation. The difference of transmission probabilities for singlet and triplet channels can reach a few per cent for a single quantum dot. For several quantum dots in the array due to interference effects it can reach approximately 100% for some energy intervals. For the same energy intervals the conductance of the device reaches the value ≈1 in [e2/π] units. As a result a model of the spin gun which transforms the spin-unpolarized electron beam into a completely polarized one is suggested.

Two‐body resonance scattering and annihilation of composite charged particles

A model of two‐body scattering including interaction between the external (Coulomb) and internal (e.g., quark) channels is constructed and investigated. A mathematically strict description is based on extensions theory for symmetric operators. Extra (internal) channel simulates complicated structure of charged particles and generates energy‐dependent effective interaction in the external channel. The main effects of this short‐range energy‐dependent interaction in the system of charged particles (Zel’dovich effect, appearance of resonances, relative shift formula, and so on) are studied. Both for models of zero–range and nonzero–range energy‐dependent interaction stationary scattering theory is constructed. In the frames of the same method a model of a system with extra (annihilation) scattering channel is considered.