Adjoint Interior-Point Boundary Conditions for Linear Differential Operators

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

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Spectral theory of some non-selfadjoint linear differential operators

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0019 ◽

2013 ◽

Vol 469 (2154) ◽

pp. 20130019 ◽

Cited By ~ 5

Author(s):

B. Pelloni ◽

D. A. Smith

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Unique Solution ◽

Spectral Theory ◽

Spectral Properties ◽

Differential Operators ◽

Boundary Value ◽

Bounded Interval ◽

Linear Differential Operators ◽

Linear Differential

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂ t q ±i Sq =0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

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Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.22.202001.48-70 ◽

2020 ◽

Vol 22 (1) ◽

pp. 48-70

Author(s):

Sergey I. Mitrokhin

Keyword(s):

Asymptotic Behavior ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Differential Operators ◽

Boundary Value ◽

Weight Functions ◽

Order Differential Operator ◽

Eighth Order ◽

Entire Family ◽

Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

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