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The aim of this lecture is to present a short summary of some aspects a mathematical construction developed at Leningrad University for two-body exactly solvable models with point interactions [1–3, 5–9].
We give a precise sense to the notion of singular perturbation. It is a bilinear form b in a Hilbert space H with a regular (closable) component br = o. Further we propose a classification of singular bilinear forms with respect to a fixed selfadjoint operator A ⩾ o in H. Finally we present a construction of the singularly perturbed operator Ab. Our definition of Ab is based on the interpretation...
We construct covariant random vector fields over 4-dimensional space-time as solutions of a system of first order coupled stochastic partial differential equations, best interpreted as equations for quaternionic valued random fields. The fields are covariant under the proper Euclidean transformations. We give necessary and sufficient conditions in terms of a given source of the infinitely divisible...
We give a short report on work done in recent years on solvable models for quantum mechanical crystals (crystals with point interactions, thus three dimensional extensions of Kronig Penney's model). We discuss the mathematical definition of the Hamiltonian and its spectral properties in the case of perfect crystals, as well as in the case of crystals with deterministic or randomly distributed point...
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 modified Faddeev equations are derived both in external and internal channels. We prove the fredholmity...
A new version of the resonating-group model with extended relative-motion space which takes into account the effect of additional two-body resonant channels is formulated using the theory of self-adjoint extensions. It is shown that after projecting on the original relative-motion space we get an effective non-local energy-dependent interaction which describes Pauli repulsion for small intercluster...
The problem of energy spectrum of a few quasi-particles in a crystal is investigated. The form of the N-magnon Hamiltonian in a Heisenberg ferromagnet is obtained and general spectral properties of the Hamiltonian as a cluster operator are demonstrated. The quasi-particle spectrum in the strong coupling limit, the Efimov effect, the current and noncurrent bound states are also discussed.
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