Search

This paper presents a proof of the Jordan-Kronecker theorem on the reduction to canonical form of a pair of skew-symmetric bilinear forms on a finite-dimensional linear space over an algebraically closed field. [ABSTRACT FROM AUTHOR]

The article presents information on operator monotone functions of several variables. The classical Löwner theorem states that a continuous function possesses the property for all pairs of bounded self-adjoint operators in the Hilbert space. The article futher offers a mathematical exposition of the theorem.

The article discusses the solvability conditions for boundary-value problems for elliptic operator-differential equations with discontinuous coefficient. It presents the results on the solvability of boundary-value problems for operator-differential equations on semiaxis. It says that the case of the coefficients in the boundary conditions as operators was not considered. It adds that only one point of discontinuity is considered for simplicity.

We consider the boundary-value problem on a finite interval for a class of second-order operator-differential equations with a linear operator in one of its boundary conditions. We obtain sufficient conditions for the regular solvability of the boundary-value problem under consideration; these conditions are expressed only in terms of its operator coefficients. [ABSTRACT FROM AUTHOR]

The article focuses on the solvability of a boundary-value problems in a second-order partial differential operator equations. It discusses the formula for Hilbert (H) space, A = U C, where A is a normal invertible operator for A and U as a unitary operator and C as a positive definite self-adjoint operator in H. It also include several theorems and operations in solving the problem using the formula mentioned above.

We supplement and further develop well-known results due to Shtraus on the generalized resolvents and spectral functions of the minimal operator L generated by a formally selfadjoint differential expression of even order with operator coefficients given on the interval [0, b〉, where b≤∞. Our approach is based on the notion of a disintegrating boundary triple, which allows us to establish a relation between the Shtraus method and boundary-value problems with spectral parameter in the boundary condition. In particular, we obtain a parametrization of all the characteristic matrices Ω( λ) of the operator L in terms of the spectral parameter corresponding to a boundary-value problem. Such a parametrization is given as a block representation of the matrix Ω( λ), as well as by formulas similar to Krein's well-known formula for generalized resolvents. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability. [ABSTRACT FROM AUTHOR]

The article discusses the properties and structure of solutions of integro-differential equations arising in thermal physics and acoustics. It says that the integro-differential equations describes heat propagation in media with sound and memory propagation in viscoelatic media. It states that representation of solutions of Sobolev spaces through series in elementary function was not considered and the series are found through the application of inverse Laplace transform to the solution problem.

The article presents the theory of radial positive definite functions on the study of spectral properties of self-adjoint extensions. It provides the classical Schoenberg theorem of a function and equation wherein its value is thought to be strictly positive definite in an inequality function. Another theorem is provided which is expressed in a nonnegative finite Borel measure, having the Bessel function of order.

We study the spectral problem related to the description of small transverse vibrations of homogeneous visco-elastic rods. The left end of the rod is hinged at the joint. The right end is attached to a concentratedmass. The spectrumof this problem is described and asymptotic formulas for eigenvalues are obtained. [ABSTRACT FROM AUTHOR]

In the space of functions with values in Hilbert space, we consider the Cauchy problem u′ + Au + B( u, u) = f( t), u(0) = 0, 0 ≤ t ≤ T. We construct examples of a self-adjoint operator A ≥ E and a bilinear transformation B satisfying the condition 〈 B( u, v), v〉 = 0 such that the Cauchy problem is not strongly solvable. [ABSTRACT FROM AUTHOR]

The article presents a study on the use of ℋ in the Hilbert space over the field ℂ, ℬ (ℋ) of all enclosed linear operators in ℋ wherein E is the identity operator in ℋ. It says that ℋ is describable in terms of annihilation and creation operators with the aid of polynomials. It adds that the lemma of Berezin uses a method with a transformer on Hilbert—Schmidt operators in which the operator C is enclosed for the generalization of the lemma.

The article discusses the boundary-value problem, considering the operator-differential equations of third order. It states that through the boundary-value problem, many nonclassical problems of mathematical physics can be simplified intto special cases of problems. It says that stating the main theorem on the regular solvability of problem can be done through theorem on intermediate derivatives. It adds that the numbers in Theorem 1 are estimates of the intermediate derivative operators' norms.

The article presents the regular solvability of the coefficients of equations with complete elementary solutions in a class of second-order differential equations and operator coefficients. It notes that the theorems indicate the exact values of the norms of intermediate derivative operators of mathematical interest in different applications and different domains of mathematical analysis.

It is proved that the D’Alembert operator in ℝ n with multidimensional time, bordered by operators of multiplication by some function, and subject to an acceptance condition at infinity is a self-adjoint operator with discrete spectrum. The spectrum and eigenfunctions are explicitly described. [ABSTRACT FROM AUTHOR]

In the present paper, we study the problem of transferring a system from one state into another state in the presence of a player trying to prevent the occurrence of this transfer in the system. The system dynamics is described by partial differential equations whose right-hand sides contain the player controls in additive form. In a similar setting, the problem was solved in several papers, but it has not been considered for the case in which various constraints are imposed on the controls of the players. Here, in contrast to several other papers, we consider games in the entire scale of spaces H r , r ≥ 0. We propose a new approach for completing the pursuit under various constraints on the player controls. [ABSTRACT FROM AUTHOR]

The article features the number of eigenvalues for a Gibbs ensemble of self-adjoint operators in the U.S. If a self-adjoint operator with discrete positive spectrum, the eigenvalues and projection operators on the corresponding eigensubspaces in the Hilbert space L2(RD), where D is any finite integer. Assuming that the number of eigenvalues are less than a given value E as E→∞ satisfied Weyl's rule.

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients. [ABSTRACT FROM AUTHOR]

We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues. [ABSTRACT FROM AUTHOR]

The article presents some problems of approximation of the generalised carathéodory function in a special domain. It cites that a function f is called a generalized carathéodory function if it is meromorphic in the open unit disk D and the kernel has a finite number of negative squares.

The article presents some problems of approximation of the generalised carathéodory function in a special domain. It cites that a function f is called a generalized carathéodory function if it is meromorphic in the open unit disk D and the kernel has a finite number of negative squares.

Presents a research on the similarity between J-Self-Adjoint Sturm-Liouville operators with operator potential and self-adjoint operators by I.M. Karabash and S. Hassi. Support of the Finish Academy of Sciences on the study; Mathematical theorems and functions used in the study; Inclusion of Banach space, Hilbert space, Riesz basis, discrete spectrum and resolvent similarity in the research.

Studies the similarity of indefinite Sturm-Liouville operators with singular potential to a selfadjoint operator. Supposition of the space functions; Identification of the natural domain of the operator; Presentation of spectral problems of the form.

In this paper, we obtain sufficient conditions for the existence of a unique regular solution of the boundary-value problems for operator differential equations of order $2k$ with variable coefficients. These conditions are expressed solely in terms of operator coefficients of the equations under study. [ABSTRACT FROM AUTHOR]

We consider an operator function $F$ defined on the interval $[\backslash sigma,\backslash tau]\backslash subset\backslash Bbb\; R$ whose values are semibounded self-adjoint operators in the Hilbert space $\backslash frak\; H$. To the operator function $F$ we assign quantities $\backslash Cal\; N\_F$ and $\backslash nu\_F(\backslash lambda)$ that are, respectively, the number of eigenvalues of the operator function $F$ on the half-interval $[\backslash sigma,\backslash tau)$ and the number of negative eigenvalues of the operator $F(\backslash lambda)$ for an arbitrary $\backslash lambda\backslash in[\backslash sigma,\backslash tau]$. We present conditions under which the estimate $\backslash Cal\; N\_F\backslash geq\backslash nu\_F(\backslash tau)-\backslash nu\_F(\backslash sigma)$ holds. We also establish conditions for the relation $\backslash Cal\; N\_F=\backslash nu\_F(\backslash tau)-\backslash nu\_F(\backslash sigma)$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider equations of the form $\backslash ddot\; x+B\backslash dot\; x+Ax=\backslash nomathbreak\; 0$, where $x=x(t)$ is a function with values in the Hilbert space $\backslash Cal\; H$, the operator $B$ is symmetric, and the operator $A$ is uniformly positive and self-adjoint in $\backslash Cal\; H$. The linear operator $\backslash Cal\; T$ generating the $C\_0$-semigroup in the energy space $\backslash Cal\; H\_1\backslash times\; \backslash Cal\; H$ is associated with this equation. We prove that this semigroup is exponentially stable if the operator $B$ is uniformly positive and the operator $A$ dominates $B$ in the sense of quadratic forms. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation of solutions of a special nonlinear operator equation used in mathematical physics is considered. In the case of a! equation for which the Fréchet derivative of the associated operator is a locally perturbed Fredholm operator, sufficient conditions for branching of solutions are studied. The methodology of application of the formalism developed in the paper is demonstrated by the example of the Boltzmann equation. [ABSTRACT FROM AUTHOR]

Considers the evolution of quantum systems with quadratic Hamiltonians in the Wiener-Segal-Fock representation. Transition from pure to mixed states; Decoherence mechanism; Hilbert space; Self-adjoint operator.

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form $$L = - \frac{{{\text{sign }}x}}{{|x|^\alpha p(x)}}\frac{{d^2 }}{{dx^2 }},{\text{ }}\alpha >- 1$$ , in the space $$L_2 (\mathbb{R})$$ with weight $$|x|^\alpha p(x)$$ . As is well known, the answer to this problem in the case $$p(x) \equiv 1$$ is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions $$p(x)$$ . [ABSTRACT FROM AUTHOR]

A second-order differential operator self-adjoint with respect to an indefinite metric on the circle is considered. The spectral resolution of this operator is found. The hypergeometric function is used in the computation of the Plancherel measure. [ABSTRACT FROM AUTHOR]

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution in L2( X, μ)-space and a convolution-type operator in L2(ℝ n)-spaces. Symmetric convolution and multiplication operators are considered in L2( X, μ) and L2(ℝ n)-spaces. [ABSTRACT FROM AUTHOR]

The Laguerre-Sonin polynomials L are orthogonal in linear spaces with indefinite inner product if α<−1. We construct the completion Π( α) of this space and describe self-adjoint extensions of the Laguerre operator l(y)=xy″+(1+ α−x) y′, α<−1, in the space Π( α). In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctions coincide with the Laguerre-Sonin polynomials and form an orthogonal basis in Π( α). [ABSTRACT FROM AUTHOR]

The article focuses on non-self-adjointness of operators with small diffusion. The question of whether the constructed (formal) asymptotic eigenfunctions and eigenvalues approximate some exact eigenfunctions and eigenvalues of the operator L turns out to be very nontrivial, and the answer to this question is still unknown. A similar question has been discussed for numerous problems of quantum and wave mechanics.