On the Ratio of Consecutive Eigenvalues

Payne, L. E.; Pólya, George; Weinberger, Hans F.

doi:10.1002/sapm1956351289

Cited by 267 publications

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“…There has been much work dedicated to extending and strengthening the classical gap inequality of Payne, Pólya, and Weinberger [28], [29] (see also [2], [4], [5], [6], [7], [34]). This result states that…”

Section: Introductionmentioning

confidence: 99%

A unified approach to universal inequalities for eigenvalues of elliptic operators

Ashbaugh

Hermi²

2004

Pacific J. Math.

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We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

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Section: Introductionmentioning

confidence: 99%

A unified approach to universal inequalities for eigenvalues of elliptic operators

Ashbaugh

Hermi²

2004

Pacific J. Math.

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“…But this means by our above concavity argument that h ′ (r) is decreasing and thus h ′ (r) < 0 for all r >r. Then Z ′ q is strictly decreasing for r ≥r. Together with Zq(r 2 ) ≤ 0 and Z ′ q (r) ≤ 0 this implies that Zq(r 3 ) < 0, a contradiction to (17). Proof.…”

Section: A Monotonicity Lemmamentioning

confidence: 88%

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Benguria

Linde

2006

Commun. Math. Phys.

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Abstract. Let λ i (Ω, V ) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ R n and with the positive potential V . Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V ⋆ , we prove that λ 2 (Ω, V ) ≤ λ 2 (S 1 , V ⋆ ). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V ) = λ 1 (S 1 , V ⋆ ).Further we prove under the same convexity assumptions on a spherically symmetric potential V , that λ 2 (B R , V )/λ 1 (B R , V ) decreases when the radius R of the ball B R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.

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“…When M = R n , for the clamped plate problem, Payne, Pólya and Weinberger [14] and [15] established a universal inequality for eigenvalues. They obtained…”

Section: Introductionmentioning

confidence: 99%

Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space

Cheng

Yang

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional hyperbolic space. It is well known that, for a bounded domain in the n-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yangtype universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the n-dimensional Euclidean space, which is sharper than the other ones. For a domain in a unit sphere, Wang and Xia (2007) have also given a universal inequality for eigenvalues. For a bounded domain in the ndimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing "nice" trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to 16.

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On the Ratio of Consecutive Eigenvalues

Cited by 267 publications

References 0 publications

A unified approach to universal inequalities for eigenvalues of elliptic operators

A unified approach to universal inequalities for eigenvalues of elliptic operators

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space

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