Quantum volterra model and the universalR-matrix

We prove that an integrable system solved by the quantum inverse scattering method can be described by a purely algebraic object (universal R-matrix) and a proper algebraic representation. For the quantum Volterra model, we construct the L-operator and the fundamental R-matrix from the universal R-matrix for the quantum affine algebra U_q(ŝl₂) and the q-oscillator representation for it. Thus, there is an equivalence between an integrable system with the symmetry algebra A and the representation of this algebra. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 384–396, December, 1997.     

Defining relations on the Hamiltonians of <Emphasis Type="Italic">XXX</Emphasis> and <Emphasis Type="Italic">XXZ R</Emphasis>-matrices and new integrable spin-orbital chains

We present several complete systems of integrability conditions on the density of the Hamiltonian of a spin chain matrix. The corresponding formulas for R-matrices are also given. The latter are expressed via the local Hamiltonian density in a form similar to spin one half XXX and XXZ models. The result is applied to the problem of integrability of SU(2) × SU(2)-and SU(2) × U(1)-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). Eight new integrable cases are found. One of these cases corresponds to the Temperley-Lieb algebra, three cases correspond to the algebra associated with the XXX model, one case corresponds to the algebra associated with the XXZ model, and one case corresponds to the algebra associated with the graded XXZ model. The remaining two R-matrices are also presented. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 50–58.     

CMV matrices with asymptotically constant coefficients. Szegö-Blaschke class, scattering theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrices acting in l², whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in l²) behaves asymptotically as the basis associated with the free system. (2) From the point of view of the scattering problem, the most natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely and (b) the associated Gelfand-Levitan-Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an A₂ kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions close to the optimal ones are given directly in terms of the scattering data.     

Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.      

Classification and properties of the spectrum of additive operators

For the algebra of operators that are additive and continuous on all of a Hilbert space we introduce a classification of the spectrum corresponding to the usual one in the case of the algebra of linear operators. We study properties of the components of the spectrum. We study the properties of the spectrum of normal self-adjoint and unitary conjugate-linear operators. The classification introduced here can be modified for closed densely defined additive operators. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 123–127.     

From Generalized Clifford Algebras to nambu’s formulation of dynamics

We consider a commutative part of the Generalized Clifford Algebras, denominated asalgebra of multicomplex numbers. By using the multicomplex algebra as underlying algebraic structure we construct oscillator model for the Nambu’s formulation of dynamics. We propose a new dynamicals principle which gives rise to two kinds of Hamilton-Nambu equations inD≥2-dimensional phase space. The first one is formulated with (D−1)-evolution parameter and a single Hamiltonian. The Haniltonian of the oscillator model in such dynamics is given byD-degree homogeneous form. In the second formulation, vice versa, the evolution of the system along a single evolution parameter is generated by (D−1) Hamiltonian.     

Applications of deformed oscillators

Some variants of the definitions and properties of deformed oscillators are reviewed. The q-analogs of coherent states are discussed. We also consider some applications of deformed oscillators, including q-oscillator representations of the simplest quantum algebras and superalgebras, q-coherent states of the su_q(2) and su_q(1,1) quantum algebras of the Jaynes-Cummings model. Generalizations are given for the case of several degrees of freedom.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 37–74, 1991.     

Spectral analysis of a two body problem with zero-range perturbation

We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one, two and three and we study its spectrum. In fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian.     

Dynamic systems related to the Cremmer-GervaisR-matrix

The generalized Cremmer-Gervais R-matrix, which is a twist of the standard sl_q (3) R-matrix, depends on two additional parameters. We discuss the properties of this R-matrix and construct two associated dynamic systems: the q-oscillator that is covariant with respect to the corresponding quantum group and an integrable spin chain with a non-Hermitian Hamiltonian. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116. No.1, pp. 101–112, July, 1998     

On Dynamical Properties of a One-Parameter Family of Transformations Arising in Averaging of Semigroups

To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.     

The q-deformed harmonic oscillator, coherent states, and the uncertainty relation

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the “coordinate-momentum” uncertainties in q-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the q-deformation results in a violation of the standard uncertainty relation; the product of the coordinate-and momentum-operator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of λ near the convergence radius of the q-exponential. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 315–322, May, 2006.     

Hamiltonian systems with a stochastic force:nonlinear versus linear,and a girsanov formula

We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process     

Integrable equations,related with the Poisson algebra

The general r-matrix scheme for the construction of integrable Hamiltonian systems is applied to a Poisson algebra, i.e., the algebra of functions on a symplectic manifold, the commutator in which is defined by the Poisson bracket. Integrable systems of two types are constructed: Hamiltonian systems on the group of symplectic diffeomorphisms, whose Hamiltonians are sums of a leftinvariant kinetic energy and a potential, and first-order systems of equations for two functions of one variable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 44–50, 1988.     

Long periodic orbits near an equilibrium and the averaging method in higher-order resonances

We consider the bifurcation of periodic orbits from an equilibrium in Hamiltonian systems. The averaging method is developed in higher-order resonance cases. For systems with general degrees of freedom, the conditions for the existence of long periodic orbits can be written in a simple form in terms of the coefficients of higher-order terms of the normalized Hamiltonian function.     

q-Deformed Euclidean algebras and their representations

A new q-deformed Euclidean algebra U_q (iso _n ), based on a definition of the algebra U_q (so _n ) different from the Drinfeld-Jimbo definition, is given. Infinite-dimensional representations T_a of this algebra, characterized by one complex number, is described. Explicit formulas for operators of these representations in an orthonormal basis are derived. The spectrum of the operator T_a(I_n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis is given. Contrary to the case of the classical Euclidean algebraiso _n, this spectrum is discrete and the spectrum points have one point of accumulation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 467–475, June, 1995.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

Representations of the deformed oscillator algebra for different choices of generators

For the q-deformed boson oscillator algebra, different commutation relations corresponding, to different choices of generators are considered in the general context. By the standard technique, the problems of construction and classification of the 1-deformed boson oscillator algebra representations in different bases are solved. The possibility of introducing the Hopf algebra structure in the q-deformed oscillator algebra is considered. The possibility of constructing the universal R-matrix for the algebra under consideration is briefly discussed. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp 80–106.     

Schauder estimates for higher-order parabolic systems with time irregular coefficients

We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and a half space. We also provide an existence result for the divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables.     

Systems of <Emphasis Type="Italic">sl</Emphasis>(2, ℂ) tops as two-particle systems

We show that Euler-Arnold tops on the algebra sl(2, ℂ) are equivalent to a two-particle system of Calogero type. We show that an arbitrary quadratic Hamiltonian of an sl(2, ℂ) top can be reduced to one of the three canonical Hamiltonians using the automorphism group of the algebra. For each canonical Hamiltonian, we obtain the corresponding two-particle system and write the bosonization formulas for the coadjoint orbits explicitly. We discuss the relation of the obtained formulas to nondynamical Antonov-Zabrodin-Hasegawa R-matrices for Calogero-Sutherland systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 8–21, October, 2008.     

Jordanian deformation of the open XXX spin chain

We find the general solution of the reflection equation associated with the Jordanian deformation of the SL(2)-invariant Yang R-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same K-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.