Operators In Spaces Of Functions Research Articles

Pseudodifferential Operators in Function Spaces with Exponential Weights

This paper is a continuation of [17]. We study weighted function spaces of type Bapq(u) and Fapq(u) on the Euclidean space ℝn, where u is a weight function of at most exponential growth. In particular, u(x) = exp(±|x|) is an admissible weight. We consider symbols which belong to the Hormander class Su1,δ, where u ∈ ℝ and 0 ≤ δ ≤ 1. We give sufficient conditions for the boundedness of the corresponding pseudodifferential operators in the above function spaces. As a main tool, we use molecular decompositions of these spaces. Furthermore, we prove that the spaces Bapq (u) and Fapq(u) have the lift property.

Bounded and Almost Periodic Solutions of Linear High-Order Hyperbolic Equations

The present paper is devoted to investigating high-order strictly hyperbolic operators with nearly constant coefficients. The invertibility of these operators in spaces of functions uniformly bounded or almost periodic in time is established, provided the symbol of the operator in question has no roots in an open strip containing the real line. Under the additional condition that the strip coincides with a half-plane, exponentially decaying solutions of the nonhomogeneous equation with right-hand side of exponential decay are constructed. In the case of equations with constant coef- ficients the necessity of the derived conditions is proved. The results of this paper were announced without proofs in (SV).

NONTRIVIAL EXPANSIONS OF ZERO IN ABSOLUTELY REPRESENTING SYSTEMS. APPLICATIONS TO CONVOLUTION OPERATORS

By using a general representation of nontrivial expansions of zero in absolutely representing systems of the form , where , is the Mittag-Leffler function, and are complex numbers, the author obtains a number of results in the theory of -convolution operators in spaces of functions that are analytic in -convex domains (a description of the general solution of a homogeneous -convolution equation and of systems of such equations, a topological description of the kernel of a -convolution operator, the construction of principal solutions, and a criterion for factorization).

Volterra integral operators in spaces of functions of two variables

The solvability in the spaces C and Lp, 1⩽p⩽ ∞, of the linear equation $$\lambda x(t,s) = Ax(t,s) + f(t,s),$$ , where $$Ax(t,s) = c(t,s)x(t,s) + \int\limits_0^t {m(t,\tau ,s)x(\tau ,s)d\tau + } \int\limits_0^s n (t,s,\sigma )x(t,\sigma )d\sigma + \int\limits_0^t {\int\limits_0^s {k(t,\tau ,s,\sigma )x(\tau ,\sigma )d\tau d\sigma .} } $$ is investigated.

Normal solvability and the noethericity of elliptic operators in spaces of functions on Rn. Part II

Normal solvability and the noethericity of elliptic operators in spaces of functions on Rn. Part II

Normal solvability and the noetherictty of elliptic operators in spaces of functions on Rn, part I

One considers differential operators, elliptic in the Petrovskii sense, whose coefficients are defined and bounded in the entire space ℝn. One finds necessary and sufficient conditions for their Noethericity and normal solvability with a finite-dimensional kernel in Holder spaces.

Asymptotic distribution of eigenvalues of secondarily quantized operator

Introduction. Considerable advances have recently been made in the problem of the asymptotic behavior of the discrete spectrum of a pseudodifferential operator (see survey in [I]). The available results refer to operators in spaces of functions of a finite number of arguments, and include results in the problem of quantum mechanics concerning the asymptotic behavior of the energy levels as the number increases.

Operators in function spaces which commute with multiplications

Operators in function spaces which commute with multiplications

Interpolation theorems for operators in function spaces

Interpolation theorems for operators in function spaces

Quantum Mechanics of Beats between Weakly Coupled Oscillators

Although the weakly coupled double pendulum seems to be a standard illustration in classical mechanics, it is seldom mentioned in quantum mechanics. It shares this fate with the damped harmonic oscillator, but while there is a good reason for avoiding the last one in quantum mechanics, the first can be treated with the same procedure as in classical mechanics. The exact solution is then compared with the solutions obtained by time-independent and time-dependent perturbation methods. It turns out that there are some additional steps to be taken, compared to the usual textbook treatment of the time-independent perturbation theory, due to the fact that all the levels of the unperturbed problem are degenerate. It is shown that the diagonalization of the secular matrix in the time-independent problem is equivalent to a problem of rotation of angular momentum operators in function space. The time-dependent problem needs, again, additional steps due to the degeneracy, and it yields the well-known beat-behavior.