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

scholarly journals On some connection results between Laguerre polynomials via third-order differential operator

2020 ◽  
Author(s):  
Baghdadi Aloui ◽  
wathek chammam ◽  
Jihad Souissi
Keyword(s):  
Differential Operator ◽  
Laguerre Polynomials ◽  
Order Differential Operator ◽  
Third Order ◽  
Orthogonal Sequence

Let $\{L^{(\alpha)}_n\}_{n\geq 0}$, ($\alpha\neq-m, \ m\geq1$), be the monic orthogonal sequence of Laguerre polynomials. We give a new differential operator, denoted here $\mathscr{L}^{+}_{\alpha}$, raises the degree and also the parameter of $L^{(\alpha)}_n(x)$. More precisely, $\mathscr{L}^{+}_{\alpha}L^{(\alpha)}_n(x)=L^{(\alpha+1)}_{n+1}(x), \ n\geq0$. As an illustration, we give some properties related to this operator and some other operators in the literature, then we give some connection results between Laguerre polynomials via this new operator.

25.—Spectrum of a Third-order Differential Operator with Large Coefficients

1975 ◽  
Vol 72 (4) ◽  
pp. 299-305
Author(s):  
K. Unsworth
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Order Differential Operator ◽  
Third Order ◽  
Minimal Operator ◽  
The Third ◽  
Entire Real Axis

SynopsisThis paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

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.

Self-adjoint extensions of singular third-order differential operator and applications

2014 ◽  
Author(s):  
Elgiz Bairamov ◽  
Meltem Sertbaş ◽  
Zameddin I. Ismailov
Keyword(s):  
Differential Operator ◽  
Order Differential Operator ◽  
Third Order

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.

Sign in / Sign up

Export Citation Format

Share Document