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 the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

The least limit point of the spectrum associated with sinǵular differential operators

1970 ◽  
Vol 67 (2) ◽  
pp. 277-281 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Operator ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Limit Point ◽  
Compact Support ◽  
Complex Space ◽  
Differential Operators ◽  
Adjoint Extension ◽  
Image Position

Let τ be the formally self-adjoint differential operator denned bywhere the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

Essential Spectra of General Second-Order Differential Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Essential Spectrum ◽  
Distance Function ◽  
Differential Operators ◽  
Second Order ◽  
Compact Perturbation ◽  
Essential Spectra ◽  
Sectorial Operators

In this chapter, the operators considered are those m-sectorial operators discussed in Chapter VII, and the essential spectra are the sets defined in Chapter IX that remain invariant under compact perturbation. A generalization of a result of Persson is used to determine the least point of the essential spectrum. Davies’ mean distance function is introduced and consequences investigated.

scholarly journals Limit circle criteria for 2nth order differential operators

1981 ◽  
Vol 24 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ronald I. Becker
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Limit Circle ◽  
All Solutions ◽  
Order Case ◽  
Order Coefficient ◽  
Image Position

A formally self-adjoint differential operatorLis said to be of limit circle type at infinity if its highest order coefficient is zero-free and all solutionsxofL(x) = 0 are square-integrable on [a, ∞). (We will drop reference to “at infinity” in what follows.)For the second-order caseDunford and Schwartz (3) p. 1409 prove thatgiventhenLis of limit circle type if and only if

Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators

1969 ◽  
Vol 21 ◽  
pp. 460-465
Author(s):  
Kurt Kreith
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Lower Bounds ◽  
Fourth Order ◽  
Differential Operators ◽  
Intimate Connection ◽  
Sturm Liouville ◽  
Image Position ◽  
Oscillation Properties ◽  
Liouville Operators

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

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.

On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators

1982 ◽  
Vol 92 (3-4) ◽  
pp. 271-300
Author(s):  
F. V. Atkinson ◽  
W. D. Evans
Keyword(s):  
Differential Equation ◽  
Linear Operator ◽  
Essential Spectrum ◽  
General Result ◽  
Differential Operators ◽  
Exponential Behaviour ◽  
Image Position ◽  
Complex Valued

SynopsisThe paper deals with the differential equationon [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r<1.

scholarly journals AdS superprojectors

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
E. I. Buchbinder ◽  
D. Hutchings ◽  
S. M. Kuzenko ◽  
M. Ponds
Keyword(s):  
Shell Model ◽  
Differential Operators ◽  
Second Order ◽  
Projection Operators ◽  
De Sitter ◽  
Gauge Invariant ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Higher Spin

Abstract Within the framework of $$ \mathcal{N} $$ N = 1 anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)}:= {\mathfrak{V}}_{\left(\alpha 1\dots \alpha m\right)\left({\overset{\cdot }{\alpha}}_1\dots {\overset{\cdot }{\alpha}}_n\right)} $$ V α m α ⋅ n ≔ V α 1 … αm α ⋅ 1 … α ⋅ n on AdS superspace, with m and n non-negative integers, the corresponding superprojector turns $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)} $$ V α m α ⋅ n into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gauge transformations are derived. We give a systematic discussion of how to realise the unitary and the partially massless representations of the $$ \mathcal{N} $$ N = 1 AdS4 superalgebra $$ \mathfrak{osp} $$ osp (1|4) in terms of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS4. We also prove that the gauge-invariant actions for superconformal higher-spin multiplets factorise into products of minimal second-order differential operators.

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