5.—Semi-bounded Dirichlet Integrals and the Invariance of the Essential Spectra of Self-adjoint Operators

1976 ◽  
Vol 75 (1) ◽  
pp. 41-66 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Symmetric Operator ◽  
Significant Feature ◽  
Dirichlet Integral ◽  
Sesquilinear Forms ◽  
Degenerate Elliptic ◽  
Order Degenerate ◽  
Image Position ◽  
Adjoint Operators ◽  
Dirichlet Integrals ◽  
Bounded Below

SynopsisIn the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integralassociated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

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.

Existence conditions for two-parameter eigenvalue problems

1981 ◽  
Vol 91 (1-2) ◽  
pp. 15-30 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Compact Resolvent ◽  
Existence Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the conesWe obtain results under various conditions on the Tr, but the following is typical:THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

16.—A Note on Adjoint Operators

1975 ◽  
Vol 72 (3) ◽  
pp. 215-217
Author(s):  
A. Dijksma ◽  
H. S. V. de Snoo
Keyword(s):  
Linear Operator ◽  
Linear Form ◽  
Symmetric Operator ◽  
Inner Product ◽  
Image Position ◽  
Adjoint Operators

In the Hilbertspace with (,) as inner product we consider the linear operator L with domain D(L) and the sesqui-linear form 〈, 〉 defined byLet the symmetric operator L0 be the restriction of L to D(L0), where

VIII.—Some Series and Integrals involving Associated Legendre Functions, regarded as Functions of their Degrees

1936 ◽  
Vol 55 ◽  
pp. 85-90 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Infinite Series ◽  
Contour Integration ◽  
Legendre Functions ◽  
Dirichlet Integral ◽  
Associated Legendre Functions ◽  
Image Position ◽  
Dirichlet Integrals

§ 1. Introductory—In a former paper (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144) the author discussed the evaluation of a number of integrals of Associated Legendre Functions, regarded as functions of their degrees. The methods employed depended mainly on contour integration, and most of the integrals were evaluated in terms of infinite series of Associated Legendre Functions. In the present paper the methods employed are of a more elementary character, depending mainly on the use of Dirichlet Integrals; the results obtained are more general; and the integrals and the corresponding series are evaluated in simpler forms. The same notation is employed as in the previous paper. The Mehler-Dirichlet Integralwhere o < θ < π, μ > – ½, is used throughout.

scholarly journals Elliptic multiparameter eigenvalue problems

1987 ◽  
Vol 30 (2) ◽  
pp. 215-228 ◽  
Author(s):  
P. A. Binding ◽  
K. Seddighi
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Positive Definite ◽  
Ellipticity Condition ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Compact Resolvents

We study the eigenproblemwhereand Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

The Spectra of Semi-Normal Singular Integral Operators

1970 ◽  
Vol 22 (1) ◽  
pp. 134-150 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Principal Value ◽  
Image Position ◽  
Adjoint Operators

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

On the Right Hamiltonian for Singular Perturbations: General Theory

1997 ◽  
Vol 09 (05) ◽  
pp. 609-633 ◽  
Author(s):  
Hagen Neidhardt ◽  
Valentin Zagrebnov
Keyword(s):  
General Theory ◽  
Symmetric Operator ◽  
Stability Domain ◽  
Bounded Operators ◽  
Friedrichs Extension ◽  
Abstract Problem ◽  
Dense Domain ◽  
Adjoint Extension ◽  
The Right ◽  
Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

2006 ◽  
Vol 136 (4) ◽  
pp. 701-708 ◽  
Author(s):  
Jonathan Bevan ◽  
Pablo Pedregal
Keyword(s):  
Lower Semicontinuous ◽  
Short Note ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Lower Semicontinuous Envelope ◽  
Variational Integrals ◽  
Non Local ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Bounded Below

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

Sign in / Sign up

Export Citation Format

Share Document