The essential self-adjointness of differential operators

Keller, R

doi:10.1017/s0308210500011264

Cited by 6 publications

(10 citation statements)

References 11 publications

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“…We define j , T o and T as in [5] with q(x) being defined by: (iv) will be assumed to hold in any lemma requiring X o to be e.s.a.…”

Section: Ru = (-L) M L 2m U + X (-Mmentioning

confidence: 99%

“…In an Addendum (both to [5] and to this paper) we show how further flexibility can be given to the coefficient function q, in the manner of Kato and Eastham, Evans and McLeod.…”

mentioning

confidence: 94%

“…We shall use many results from our previous paper [5, in fact all of § §1-3 and much of §4 and §5] and number the additional results that we need here as extensions to that paper, certain key results being restated without proof. We refer the reader to [5] for the proofs, and also for the notation we use.…”

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confidence: 99%

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The essential self-adjointness of differential operators with positive coefficients

Keller

1979

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and pt ≧0 for t ≧1 and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions (including the positivity of the coefficients) have to be imposed only in an increasing sequence of annular regions surrounding the origin.

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“…We define j , T o and T as in [5] with q(x) being defined by: (iv) will be assumed to hold in any lemma requiring X o to be e.s.a.…”

Section: Ru = (-L) M L 2m U + X (-Mmentioning

confidence: 99%

“…In an Addendum (both to [5] and to this paper) we show how further flexibility can be given to the coefficient function q, in the manner of Kato and Eastham, Evans and McLeod.…”

mentioning

confidence: 94%

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The essential self-adjointness of differential operators with positive coefficients

Keller

1979

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Over many decades now, various sufficient conditions for the (essential) self‐adjointess of Schrödinger operators have been established; see the books for a discussion of related results. Parallel to numerous developments in the realm of Schrödinger operators, past few decades have witnessed quite a bit of activity concerning the self‐adjointness of higher order operators in

L^{2} false(R^{n} false)

; see, for instance, and references therein.…”

Section: Introductionmentioning

confidence: 99%

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

Milatovic

2017

Mathematische Nachrichten

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We give a sufficient condition for the essential self-adjointness of a perturbed biharmonic-type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions. K E Y W O R D SBiharmonic operator, Bochner Laplacian, perturbation, Riemannian manifold, self-adjointness M S C ( 2 0 1 0 ) 35P05, 47B25, 58J05, 58J50 INTRODUCTIONThe question of self-adjointness of differential operators in 2 (ℝ ) has been in the focus of attention of researchers for a long time. In particular, Schrödinger operators have occupied a special place because of their importance in mathematical physics. Over many decades now, various sufficient conditions for the (essential) self-adjointess of Schrödinger operators have been established; see the books [12,22,31] for a discussion of related results. Parallel to numerous developments in the realm of Schrödinger operators, past few decades have witnessed quite a bit of activity concerning the self-adjointness of higher order operators in 2 (ℝ ); see, for instance, [9,23,24,27] and references therein.The study of self-adjointness of differential operators on Riemannian manifolds was initiated in the landmark paper [14]. Subsequently, the authors of [10] and [11] established the self-adjointness of positive integer powers of the scalar Laplacian (and Laplacian on differential forms). In the last two decades, there has been a lot of activity on the problem of self-adjointness of Schrödinger operators on Riemannian manifolds (including operators acting on sections of Hermitian vector bundles), as seen, for instance, in [2,[6][7][8]16,17,20,25,29,30,32]. Recently, the authors of [3] proved, among other things, the essential selfadjointness of powers of first-order elliptic operators with low-regularity coefficients on vector bundles with low-regularity coefficient metrics over manifolds with low-regularity metrics.Our paper is concerned with the (essential) self-adjointness of ( ∇ † ∇ ) 2 + , the square of the Bochner Laplacian plus a potential ; see Section 2.1 for an explanation of notations. Methodologically, our problem is handled similarly as in [27], with modifications in some estimates due to the geometry of the manifold and nature of our operator. To help us with our estimates, we use a family of cut-off functions constructed recently by the authors of [4] in the context of a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a certain non-positive function; see the assumption (R) in Section 2 below

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“…The ring conditions have also been studied by a school of other mathematicians whose names include Devinatz, Eastham, Everitt, Keller, Knowles, McLeod (see for example [8,9,10,11,13], [12] and its bibliography). In [12], Keller considered a special type of 2m-th order elliptic operator. For example, in the case m = 3, T is given by Tu = (-A) 2 u + A(p 2 Au) -V .…”

Section: Introductionmentioning

confidence: 99%