2.—Semi-bounded Second-order Differential Operators

1974 ◽  
Vol 72 (1) ◽  
pp. 9-16 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Lower Bound ◽  
Differential Expression ◽  
Differential Operators ◽  
Second Order ◽  
Image Position ◽  
Bounded Below

SynopsisDifferential operators generated by the differential expression My(x) = —y″(x)+q(x)y(x) in L2(0, ∞) are considered. It is assumed thatis bounded for all x in [0, ∞) and some fixed ω > 0. The operators are shown to be bounded below and an estimate for the lower bound is obtained in terms of q(x). In the case where q(x) is LP (0, ∞) for some p ≧ 1, the results are compared with recent ones of W. N. Everitt. Some comments are made on the best-possible nature of the results.

18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators

1976 ◽  
Vol 74 ◽  
pp. 239-252 ◽  
Author(s):  
M. S. P. Eastham ◽  
W. N. Everitt
Keyword(s):  
Essential Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Image Position ◽  
Differentiability Properties

SynopsisThe paper gives asymptotic estimates of the formas λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

On a fourth-order singular integral inequality

1978 ◽  
Vol 80 (3-4) ◽  
pp. 249-260 ◽  
Author(s):  
A. Russell
Keyword(s):  
Differential Expression ◽  
Fourth Order ◽  
Singular Integral ◽  
Integral Inequality ◽  
Second Order ◽  
New Order ◽  
Alternative Account ◽  
The Real ◽  
Image Position ◽  
Complete Solutions

SynopsisThe inequality considered in this paper iswhereNis the real-valued symmetric differential expression defined byGeneral properties of this inequality are considered which result in giving an alternative account of a previously considered inequalityto which (*) reduces in the casep=q= 0,r= 1.Inequality (*) is also an extension of the inequalityas given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

Discrete spectra criteria for singular differential operators with middle terms

1975 ◽  
Vol 77 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Adjoint Extension ◽  
Image Position ◽  
Discrete Spectra ◽  
Bounded Below

Let l be the differential operator of order 2n defined bywhere the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

Second-Order Differential Operators on Arbitrary Open Sets

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Differential Operators ◽  
Neumann Boundary ◽  
Second Order ◽  
Sesquilinear Forms ◽  
Sectorial Operators ◽  
Mechanical Interpretation ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Distributional Inequality

In this chapter, three different methods are described for obtaining nice operators generated in some L2 space by second-order differential expressions and either Dirichlet or Neumann boundary conditions. The first is based on sesquilinear forms and the determination of m-sectorial operators by Kato’s First Representation Theorem; the second produces an m-accretive realization by a technique due to Kato using his distributional inequality; the third has its roots in the work of Levinson and Titchmarsh and gives operators T that are such that iT is m-accretive. The class of such operators includes the self-adjoint operators, even ones that are not bounded below. The essential self-adjointness of Schrödinger operators whose potentials have strong local singularities are considered, and the quantum-mechanical interpretation of essential self-adjointness is discussed.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

scholarly journals On Self-Adjoint Partial Differential Equations of the Second Order

1924 ◽  
Vol 43 ◽  
pp. 35-38 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Expression ◽  
Second Order ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Differential ◽  
Image Position

Let be a linear differential expression involving n independent variables xi the coefficients AikBi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

Decaying positive solutions to a system of nonlinear elliptic equations

2006 ◽  
Vol 136 (2) ◽  
pp. 301-320
Author(s):  
Fenfei Chen ◽  
Miaoxin Yao
Keyword(s):  
Positive Solution ◽  
Elliptic System ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Second Order ◽  
Nonlinear Elliptic System ◽  
Nonlinear Elliptic ◽  
Image Position ◽  
Bounded Below

In this paper, the second-order nonlinear elliptic system with α, γ < 1 and β ≥ 1, is considered in RN, N ≥ 3. Under suitable hypotheses on functions fi, gi, hi (i = 1, 2) and P, it is shown that this system possesses an entire positive solution , 0 < θ < 1, such that both u and v are bounded below and above by constant multiples of |x|2−N for all |x| ≥ 1.

A second order Dirichlet differential expression that is not bounded below

1979 ◽  
Vol 83 (1-2) ◽  
pp. 39-43
Author(s):  
Man Kam Kwong
Keyword(s):  
Differential Expression ◽  
Dirichlet Condition ◽  
Second Order ◽  
Bounded Below

SynopsisWe give in this note a second order singular differential expression of the form Lf = −f″ + qf on [0, ∞) that satisfies the Dirichlet condition but that is not bounded below.

Dirichlet and separation results for Schrödinger-type operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 151-162 ◽  
Author(s):  
W. D. Evans ◽  
A. Zettl
Keyword(s):  
Differential Expression ◽  
Maximal Operator ◽  
Second Order ◽  
Image Position ◽  
Schrödinger Type Operators ◽  
Separation Results

SynopsisConditions are obtained which ensure that the maximal operator generated by the formally self-adjoint second-order differential expressionin L2(Rn), n ≥ 1 has the Dirichlet and separation properties.

On the location of eigenvalues of second order linear differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 15-22 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Boundary Condition ◽  
Hilbert Space ◽  
Differential Expression ◽  
Differential Operators ◽  
Homogeneous Boundary Condition ◽  
Upper Bounds ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

Sign in / Sign up

Export Citation Format

Share Document