Yu. Kh. Eshkabilov

Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order

PHASE TRANSITIONS FOR A MODEL WITH UNCOUNTABLE SET OF SPIN VALUES ON A CAYLEY TREE

k\geqslant 1

The Efimov effect for a model “three-particle” discrete Schrödinger operator

. It is known that the ‘splitting Gibbs measures’ of the model can be described by solutions of a nonlinear integral equation. For arbitrary

On infinity of the discrete spectrum of operators in the Friedrichs model

k\geqslant 2

Essential and discrete spectra of partially integral operators

we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.

Partially integral operators with bounded kernels

In this paper we consider a model with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 2. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For k = 2 and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e., there are phase transitions).

On the discrete spectrum of partial integral operators

We study the existence of an infinite number of eigenvalues for a model “three-particle” Schrödinger operator H. We prove a theorem on the necessary and sufficient conditions for the existence of an infinite number of eigenvalues of the model operator H below the lower boundary of its essential spectrum.

Spectrum of a model three-particle Schrödinger operator

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrödinger operator is described.

On the essential and the discrete spectra of a Fredholm type partial integral operator

AbstractLet Ω1, Ω2 ⊂ ℝν be compact sets. In the Hilbert space L2(Ω1 × Ω2), we study the spectral properties of selfadjoint partially integral operators T1, T2, and T1 + T2, with