Separation of bi-harmonic differential operators on Riemannian
manifolds

Atia, H.A.; Alsaedi, Ramzi; Ramady, Ahmed

doi:10.1515/forum-2013-0127

Cited by 9 publications

(10 citation statements)

References 6 publications

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“…Thus, for general geodesically complete Riemannian manifolds, the essential self‐adjointness of

true(Δ^{2} + w true) |_{C_{c}^{\infty} false(M false)}

with

1 \leq w \in L_{prefixloc}^{\infty} false(M false)

is still an open question (see Remark below for an additional comment). Finally, unlike the separation result of , where the “derivative” assumptions on w seem somewhat complicated (they involve “test functions”

u \in C_{c}^{\infty} false(M false)

and are similar in spirit to the inequalities from our Lemma below), “derivative” assumptions on w in our Corollary involve only w and a constant σ, as in separation results of .…”

Section: Introductionmentioning

confidence: 93%

“…Recently, the authors of studied the separation property of

{normalΔ}^{2} + w

on a geodesically complete Riemannian manifold M , where Δ is the Laplace–Beltrami operator,

w \in C^{2} false(M false)

, and

w \geq 1

. We note that the proof of separation in used the essential self‐adjointness of

true(Δ^{2} + w true) |_{C_{c}^{\infty} false(M false)}

, and this property, as far as we know, has not been established yet in the context of general geodesically complete Riemannian manifolds. Fortunately, our Theorem provides the required justification, but only in the case of a geodesically complete Riemannian manifold with Ricci curvature satisfying the assumption (R) described in Section below.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Thus, for general geodesically complete Riemannian manifolds, the essential self‐adjointness of

true(Δ^{2} + w true) |_{C_{c}^{\infty} false(M false)}

with

1 \leq w \in L_{prefixloc}^{\infty} false(M false)

u \in C_{c}^{\infty} false(M false)

and are similar in spirit to the inequalities from our Lemma below), “derivative” assumptions on w in our Corollary involve only w and a constant σ, as in separation results of .…”

Section: Introductionmentioning

confidence: 93%

“…Recently, the authors of studied the separation property of

{normalΔ}^{2} + w

on a geodesically complete Riemannian manifold M , where Δ is the Laplace–Beltrami operator,

w \in C^{2} false(M false)

, and

w \geq 1

. We note that the proof of separation in used the essential self‐adjointness of

true(Δ^{2} + w true) |_{C_{c}^{\infty} false(M false)}

Section: Introductionmentioning

confidence: 99%

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

Milatovic

2017

Mathematische Nachrichten

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“…We say the expression

Δ + V

is separated if

u \in L^{2} false(R^{n} false)

and

false(normalΔ + V false) u \in L^{2} false(R^{n} false)

imply

Δ u \in L^{2} false(R^{n} false)

and

V u \in L^{2} false(R^{n} false)

. We should mention that the separation problem for perturbations of the biharmonic operator has been studied, previous to Milatovic's work, by the authors of [1]. However, in that paper the authors assume that the potential w satisfies certain derivative assumptions, defined via testing it against suitable test functions

u \in C_{c}^{\infty} false(M false)

.…”

Section: Introductionmentioning

confidence: 99%

Essential self‐adjointness of perturbed quadharmonic operators on Riemannian manifolds with an application to the separation problem

Saratchandran

2021

Mathematische Nachrichten

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We consider perturbed quadharmonic operators, normalΔ4+V, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential V satisfying a bound from below by a non‐positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vector bundle and the underlying Riemannian manifold, we give a sufficient condition for the essential self‐adjointness of such operators. We then apply this to prove the separation property in L2 when the perturbed operator acts on functions.

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“…For more backgrounds concerning to our problem, see [6][7][8]. Atia et al [9] have studied the separation property of the bi-harmonic differential expression A = 2 M + q , on a Riemannian manifold (M, g) without boundary in L 2 (M) , where M is the Laplacian on M and 0 ≤ q ∈ L 2 loc (M) is a real-valued function. Recently, Atia [10] has studied the sufficient conditions for the magnetic bi-harmonic differential operator B of the form B = 2 E + q to be separated in L 2 (M) , on a complete Riemannian manifold M, g with metric g , where E is the magnetic Laplacian on M and q ≥ 0 is a locally square integrable function on M. In [11], Milatovic has studied the separation property for the Schrodinger-type expression of the form L = M +q , on noncompact manifolds in L p (M) .…”

Section: Introductionmentioning

confidence: 99%

Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds

Atia

2019

J Egypt Math Soc

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Consider the bi-harmonic differential expression of the form A = 2 M + q on a manifold of bounded geometry (M, g) with metric g, where M is the scalar Laplacian on M and q ≥ 0 is a locally integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in L p (M) , if for all u ∈ L p (M) such that Au ∈ L p (M), we have qu ∈ L p (M). In this paper, we give sufficient conditions for A to be separated in L p (M) ,where 1 < p < ∞.

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Separation of bi-harmonic differential operators on Riemannian manifolds

Cited by 9 publications

References 6 publications

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

Essential self‐adjointness of perturbed quadharmonic operators on Riemannian manifolds with an application to the separation problem

Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds

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