Deficiency indices of an odd-order differential operator

1984 ◽  
Vol 97 ◽  
pp. 223-231 ◽  
Author(s):  
R. B. Paris ◽  
A. D. Wood
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Asymptotic Solutions ◽  
Order Differential Operator ◽  
Deficiency Indices ◽  
Odd Order ◽  
Associated Operators

SynopsisWe obtain asymptotic solutions of odd-order formally self-adjoint differential equations with power coefficients and discuss possible values for the deficiency indices of the associated operators.

Asymptotic theory for third-order differential equations with extension to higher odd-order equations

1991 ◽  
Vol 117 (3-4) ◽  
pp. 215-223 ◽  
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equations ◽  
Asymptotic Theory ◽  
Order Theory ◽  
Linear Differential Equations ◽  
Asymptotic Solutions ◽  
Third Order ◽  
Odd Order ◽  
Linear Differential ◽  
General Conditions

SynopsisAn asymptotic theory is developed for linear differential equations of odd order. Theory is applied with large coefficients. The forms of the asymptotic solutions are given under general conditions on the coefficients.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

The deficiency indices of a symmetric third-order differential operator

1995 ◽  
Vol 58 (3) ◽  
pp. 1002-1005
Author(s):  
V. R. Mukimov ◽  
Ya. T. Sultanaev
Keyword(s):  
Differential Operator ◽  
Order Differential Operator ◽  
Third Order ◽  
Deficiency Indices

On the spectrum of an odd-order differential operator

2017 ◽  
Vol 101 (5-6) ◽  
pp. 755-758 ◽  
Author(s):  
A. M. Akhtyamov
Keyword(s):  
Differential Operator ◽  
Order Differential Operator ◽  
Odd Order

scholarly journals Asymptotics of the Spectrum of a Periodic Boundary ValueProblem for an Odd-Order Differential Operator

2021 ◽  
pp. 5-17
Author(s):  
Sergey Mitrokhin ◽  
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Fundamental System ◽  
Regular Polygon ◽  
Periodic Boundary Conditions ◽  
Order Differential Operator ◽  
Indicator Diagram ◽  
Odd Order ◽  
Asymptotics Of The Eigenvalues

The spectrum of a differential operator of high odd order with periodic boundary conditions is studied. The asymptotics of the fundamental system of solutions of the differential equation defining the operator are obtained by the method of successive Picard approximations. With the help of this fundamental system of solutions the periodic boundary conditions are studied. As a result, the equation for the eigenvalues of the differential operator under study is obtained, which is a quasi-polynomial. The indicator diagram of this equation, which is a regular polygon, is investigated. In each of the sectors of the complex plane, defined by the indicator diagram, the asymptotics of the eigenvalues of the operator under study is found. An equation for the eigenvalues of the differential operator under study is derived. The indicator diagram of this equation has been studied. The asymptotics of the eigenvalues of the studied operator in different sectors of the indicator diagram is found.

Asymptotic theory and deficiency indices for differential equations of odd order

1981 ◽  
Vol 90 (3-4) ◽  
pp. 263-279 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equations ◽  
Asymptotic Theory ◽  
Linear Differential Equations ◽  
Deficiency Indices ◽  
Odd Order ◽  
Linear Differential ◽  
General Conditions

SynopsisAn asymptotic theory is developed for linear differential equations of odd order. The theory is applied to the evaluation of the deficiency indicesN+andN−associated with symmetric differential expressions of odd order. General conditions on the coefficients are given under which all possible values ofN+andNsubject to |N+−N| ≦ 1 are realized.

scholarly journals On the existence of optimal control for controlled stochastic partial differential equations

1989 ◽  
Vol 115 ◽  
pp. 73-85 ◽  
Author(s):  
Noriaki Nagase
Keyword(s):  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Admissible Control ◽  
Order Differential Operator ◽  
Control Problems ◽  
Partial Differential ◽  
The Cauchy Problem ◽  
Existence Of Optimal Control

In this paper we are concerned with stochastic control problems of the following kind. Let Y(t) be a d’-dimensional Brownian motion defined on a probability space (Ω, F, Ft, P) and u(t) an admissible control. We consider the Cauchy problem of stochastic partial differential equations (SPDE in short)where L(y, u) is the 2nd order elliptic differential operator and M(y) the 1st order differential operator.

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