Basis Property Of Eigenfunctions Research Articles

NON STABILITY ON BASIS PROPERTY OF SYSTEMS ROOT VECTORS OF A LOADED MULTIPLE DIFFERENTIATION OPERATOR

In this paper we consider perturbations of a second order differential equation of the spectral problem with a loaded term, containing a value of the unknown function at the point zero, with regular, but not strongly regular boundary value conditions. Question about basis property of eigen functions and associated functions (E&AF) systems of a loaded multiple differentiation operator is studied. In the case of non-self-adjoint ordinary differential operators, the basis property of systems of eigen functions and associated functions (E&AF), in addition to the boundary value conditions, can be affected by values of coefficients of the differential operator. Moreover, it is known that the basic properties of E&AF can be changed at a small change of values of the coefficients. This fact was first noted in Il’in V.A. In this paper problem non stability on basis property of systems root vectors of a loaded multiple differentiation operator.

ON BASIS PROPERTY OF SYSTEMS ROOT VECTORS OF A LOADED MULTIPLE DIFFERENTIATION OPERATOR

In the case of non-self-adjoint ordinary differential operators, the basis property of systems of eigenfunctions and associated functions (E&AF), in addition to the boundary value conditions, can be affected by values of coefficients of the differential operator. Moreover, it is known that the basic properties of E&AF can be changed at a small change of values of the coefficients. This fact was first noted in V.A. Il’in. Ideas of V.A. Il’in for the case of non-self-adjoint perturbations of the self-adjoint periodic problem were developed in A.S. Makin where operator was changed due to perturbation of one of the boundary value conditions.In Sadybekov M.A., Imanbaev N.S., we studied another version of the non-self-adjoint perturbation of the self-adjoint periodic problem. In contrast to A.S. Makin, in Sadybekov M.A. and Imanbaev N.S. perturbation occurs due to the change in the equation, which belongs to the class of so-called loaded differential equations, where the basic properties of root functions are investigated.In this paper we consider perturbations of a second order differential equation of the spectral problem with a loaded term, containing a value of the unknown function at the point zero, with regular, but not strongly regular boundary value conditions. Question about basis property of eigenfunctions and associated functions (E&AF) systems of a loaded multiple differentiation operator is studied.

Basis Property of Eigenfunctions in Lebesgue Spaces for a Spectral Problem with a Point of Discontinuity

We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a discontinuous second-order differential operator with spectral parameter in the discontinuity (transmission) conditions. This problem arises when solving the problem on the vibrations of a loaded spring with fixed endpoints. An abstract theorem on the stability of basis properties of multiple systems in a Banach space with respect to certain transformations is proved. This theorem is used when proving theorems on the basis property of eigenfunctions of a discontinuous differential operator in the Lebesgue spaces Lp ⊕ ℂ and Lp.

The basis property of eigenfunctions in the problem of a nonhomogeneous damped string

The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator \(i A\). We prove that the set of root vectors of the operator \(A\) forms a basis of subspaces in a certain Hilbert space \(H\). Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator \(A\) is a Riesz basis for \(H\).

Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

Small motions and eigenoscillations of a “fluid–barotropic gas” hydrosystem

This paper deals the problem of small motions and eigenoscillations of the hydrodynamical system” ideal fluid–barotropic gas” with regard for gravitational and capillary forces. The spectrum structure and the basis property of eigenfunctions are studied. The variational principles for eigenvalues are presented. The theorems of strong solvability of an initial boundary-value problem and the inverse Lagrange’s theorem of stability of the hydrosystem are proved.

On the basis property of eigenfunctions of a singular second-order differential operator

We consider the first boundary value problem for a singular differential operator of second order on an interval with transmission conditions at an interior point of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L2(0, 1) and forms a Riesz basis in that space.

Estimates for the eigenfunctions of the Regge problem

the problem is considered in the reduced form (1) (2). From now on, we assume that the functions q(x) and ρ(x) are integrable on the interval [0, a] and nonnegative. The asymptotic behavior of eigenvalues, completeness properties, and the basis property of eigenfunctions for the Regge problem were studied by many authors; see [2]–[8] and the references therein. This paper contains results on the character of growth of the norms of eigenfunctions for the Regge problem in the space C[0, 1] under the following normalization condition on functions from L2[0, a]: ˆ a

On the basis property of eigenfunctions of the Frankl problem with a nonlocal oddness condition of the second kind

The eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal oddness condition of the second kind are found. The completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem are analysed.

Basis property of eigenfunctions of the generalized gasedynamic problem of Frankl with a nonlocal oddness condition

In this paper, we write eigenfunctions of the generalized gasedynamic problem of Frankl with a nonlocal oddness condition using the Bessel equation. It is proved that the eigenfunctions form a Riesz basis in L 2(D +), where D + is the elliptic part of the domain.

On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition of the second kind

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem.

Existence and basisness of eigen and adjoined elements of linear operators

In this paper we continue the study described earlier in No. 5, 2006, of Russian Mathematics (Izv. Vyssh. Uchebn. Zaved., Matematika). We establish conditions, providing the asymptotics mentioned in the cited paper. We prove the basis property of eigen functions and adjoined ones in linear problems for differential equations with deviating arguments.

On the problem of small motions and normal oscillations of a viscous fluid in a partially filled container

AbstractThe famous classical S. Krein problem of small motions and normal oscillations of a viscous fluid in a partially filled container is investigated by a new approach based on a recently developed theory of operator matrices with unbounded entries. The initial boundary value problem is reduced to a Cauchy problem in some Hilbert space. The operator matrix 𝒜 is a maximal uniformly accretive operator which is selfadjoint in this space with respect to some indefinite metric. The theorem on strong solvability of the hydrodynamic problem is proved. Further, the spectrum of normal oscillations, basis properties of eigenfunctions and other questions are studied.

A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions

Problems with a spectral parameter in equations and boundary conditions form an important part of spectral theory of linear differential operators. A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [1, 2]. Boundary value problems for ordinary differential operators with a spectral parameter in boundary conditions were considered in various settings in numerous papers [3–14]. The completeness and the basis property of eigenfunctions of boundary value problems with a spectral parameter in equations and boundary conditions were studied in more detail in [7, 8]. Consider the following boundary value problem for the Dirac system with the same spectral parameter in the equations and the boundary conditions:

On basis properties of eigenfunctions of one model problem with indefinite weight

On basis properties of eigenfunctions of one model problem with indefinite weight

Basis property of eigenfunctions of a problem on antiplanar vibrations of a cylinder with external friction

Basis property of eigenfunctions of a problem on antiplanar vibrations of a cylinder with external friction