A unified approach to universal inequalities for eigenvalues of elliptic operators

Ashbaugh, Mark S.; Hermi, Lotfi

doi:10.2140/pjm.2004.217.201

Cited by 47 publications

(64 citation statements)

References 17 publications

(40 reference statements)

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“…Commutators and their relation to eigenvalue gaps and inverse spectral problems have been studied in [1,2,12,15,16,17,19], which partly inspired this work. Here H will be defined on a hypersurface, and commutators will be used to connect eigenvalue gaps and certain other functions defined on the spectrum σ(H) to the mean curvature of the manifold M , which is found to pose tight constraints on the spectrum.…”

Section: Introductionmentioning

confidence: 98%

Commutators, Eigenvalue Gaps, and Mean Curvature in the Theory of Schrödinger Operators

Harrell

2007

Communications in Partial Differential Equations

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Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = −∆ + V (x) , acting on functions of a smooth, compactHere ∆ denotes the Laplace-Beltrami operator, and the real-valued potential-energy function V (x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed.It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature:where ν = d + 1, g is a real parameter, andwith {κ j }, j = 1, . . . , d denoting the principal curvatures of M . For instance, by Theorem 3.1 and Corollary 4.4, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using H 1/4 . The "isoperimetric" parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.

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Section: Introductionmentioning

confidence: 98%

Commutators, Eigenvalue Gaps, and Mean Curvature in the Theory of Schrödinger Operators

Harrell

2007

Communications in Partial Differential Equations

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“…(See also [39], [6], [7], [17], [22].) In the present article we put those notions together with some transform techniques in order to connect together several inequalities for spectra, which have been derived by independent methods in the past.…”

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confidence: 99%

On Riesz Means of Eigenvalues

Harrell

Hermi

2011

Communications in Partial Differential Equations

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Abstract. In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.Manuscript received December 25, 2007; revised XXX XX, XXXX.

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“…And more information about universal eigenvalue inequalities can find in [2,3,16]. It is natural to consider the estimates for eigenvalues of problem (1.1) on the other Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue Estimates for Quadratic Polynomial Operator of the Laplacian

Sun

2010

Glasgow Math. J.

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Abstract. For a bounded domainin a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator 2 − a + b of the Laplacian , where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue k+1 . As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.2010 Mathematics Subject Classification. 35P15, 58C40, 53C42.

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A unified approach to universal inequalities for eigenvalues of elliptic operators

Cited by 47 publications

References 17 publications

Commutators, Eigenvalue Gaps, and Mean Curvature in the Theory of Schrödinger Operators

Commutators, Eigenvalue Gaps, and Mean Curvature in the Theory of Schrödinger Operators

On Riesz Means of Eigenvalues

Eigenvalue Estimates for Quadratic Polynomial Operator of the Laplacian

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