A theorem on homeomorphisms for an elliptic differential expression and pseudodifferential boundary conditions

1969 ◽  
Vol 21 (2) ◽  
pp. 226-230 ◽  
Author(s):  
S. O. Rushchitskaya
Keyword(s):  
Boundary Conditions ◽  
Differential Expression ◽  
Pseudodifferential Boundary

Lower bounds for the spectrum of a second order linear differential equation with a coefficient whose negative part is p-integrable

1984 ◽  
Vol 97 ◽  
pp. 105-107
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Differential Expression ◽  
Lower Bounds ◽  
Linear Operators ◽  
Order Linear Differential Equation ◽  
Negative Part ◽  
Order Linear ◽  
Linear Differential

SynopsisWe consider ihe differential expression M[y]: = −y″ + qy on [0, ∞) where q_∈ Lp [0, ∞) for some p ≧ 1. It is known that M, together with the boundary conditions y(0) = 0 or y′(0) = 0, defines linear operators on L2 [0, ∞). We obtain lower bounds for the spectra of these operators. Our bounds depend on the Lp norm of q_ and extend results of Everitt and Veling.

scholarly journals Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Nihal Yokuş
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Complex Valued

We consider the operator generated in by the differential expression , and the boundary condition , where is a complex-valued function and , with . In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of .

Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter

1980 ◽  
Vol 86 (1-2) ◽  
pp. 1-27 ◽  
Author(s):  
Aalt Dijksma
Keyword(s):  
Boundary Conditions ◽  
Differential Expression ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Real Interval ◽  
Eigenfunction Expansions ◽  
Selfadjoint Operators ◽  
Ordinary Differential Operators ◽  
Eigenvalue Parameter ◽  
Ordinary Differential Expression

SynopsisIn provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.

scholarly journals The Krein-von Neumann extension of a regular even order quasi-differential operator

2021 ◽  
Vol 41 (6) ◽  
pp. 805-841
Author(s):  
Minsung Cho ◽  
Seth Hoisington ◽  
Roger Nichols ◽  
Brian Udall
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Maximal Operator ◽  
Friedrichs Extension ◽  
Minimal Operator ◽  
Von Neumann ◽  
Even Order

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.

Finite Dimensional Perturbations of Differential Expressions

1976 ◽  
Vol 28 (5) ◽  
pp. 1082-1104 ◽  
Author(s):  
R. R. D. Kemp ◽  
S. J. Lee
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Additional Term ◽  
Extensive Study ◽  
Basic Operator ◽  
Finite Dimensional ◽  
Dimensional Range

Operators in L2, or more generally, Lp spaces, which are generated by differential expressions, have had extensive study. More recently some authors, in particular Krall [3; 4; 5; 6; 7], Kim [2], and Krall and Brown [8], have studied operators which are generated by a differential expression plus an additional term. This additional term is of the nature of a perturbation of the differential expression by an operator with finite dimensional range. However even if the basic operator is specifically of the form of a finite dimensional perturbation of a differential operator, this is not true of the adjoint, since the boundary conditions which arise on the adjoint are not appropriate to the adjoint of the differential operator alone.

Optimal lower bounds for the spectrum of a second order linear differential equation with ap-integrable coefficient

1982 ◽  
Vol 92 (1-2) ◽  
pp. 95-101 ◽  
Author(s):  
E. J. M. Veling
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Differential Expression ◽  
Lower Bounds ◽  
The Self ◽  
Order Linear Differential Equation ◽  
Linear Differential ◽  
Adjoint Operators

SynopsisIn this note the differential expressionM[y] ≡ − y” + qy, q∈Lp(ℝ+) for some p ≧ l, is considered on [0,∞) together with the boundary condition either y(0) = 0 or y'(0) = 0. Lower bounds are given for the spectrum of the self-adjoint operatorsTgenerated by M[·] and these boundary conditions. The bounds depend on theLp-norm of the coefficientqand they improve results of Everitt and Eastham. The bounds are optimal.

Boundary conditions for general ordinary differential operators and their adjoints

1990 ◽  
Vol 114 (1-2) ◽  
pp. 99-117 ◽  
Author(s):  
W. D. Evans ◽  
Sobhy E. Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Expression ◽  
Differential Operators ◽  
Adjoint Equation ◽  
Complex Conjugation ◽  
Deficiency Indices ◽  
Ordinary Differential Operators ◽  
Adjoint Operators ◽  
Symmetric Operators ◽  
Special Case

SynopsisA characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].

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