Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator

Benguria, Rafael D.; Linde, Helmut; Loewe, Benjamin

doi:10.1007/s13373-011-0017-0

Cited by 25 publications

(18 citation statements)

References 84 publications

(105 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[7] for further references. A recent general review of isoperimetric inequalities for the Dirichlet, Neumann and other Laplacians on Euclidean spaces was made by Benguria, Linde and Loewe in [4]. We also refer to Henrot [7] and Brasco and Franzina [5] for more historic remarks on isoperimetric inequalities, namely the Rayleigh-Faber-Krahn inequality and the Hong-Krahn-Szegö inequality.…”

Section: Introductionmentioning

confidence: 99%

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

Ruzhansky

2016

Bull. Math. Sci.

View full text Add to dashboard Cite

In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere S n and on the real hyperbolic space H n . It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on H n and prove the Hong-Krahn-Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

show abstract

Section: Introductionmentioning

confidence: 99%

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

Ruzhansky

2016

Bull. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…The former impression perhaps stems from the extensive body of work in trying to just provide sharp lower and upper bounds on the first eigenvalue gap λ 2 − λ 1 under various conditions (e.g. [42,2,4,12,3]), or various other conjectured lower bounds on the entire spectrum, such as Polya's conjecture (see e.g. [28,54,48]).…”

Section: Spectrum Comparison For Positively Curved Weighted-manifoldsmentioning

confidence: 99%

Spectral estimates, contractions and hypercontractivity

Milman

2018

J. Spectr. Theory

View full text Add to dashboard Cite

Sharp comparison theorems are derived for all eigenvalues of the weighted Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to pexponential measures, unit-balls of ℓ n p , one-dimensional spaces and Riemannian submersions. Our main tool is a general Contraction Principle for "eigenvalues" on arbitrary metric-measure spaces. Motivated by Caffarelli's Contraction Theorem, we put forth several conjectures pertaining to the existence of contractions from the canonical sphere (and Gaussian space) to weighted-manifolds of appropriate topological type having (generalized) Ricci curvature positively bounded below; these conjectures are consistent with all known isoperimetric, heat-kernel and Sobolev-type properties of such spaces, and would imply sharp conjectural spectral estimates. While we do not resolve these conjectures for the individual eigenvalues, we verify their Weyl asymptotic distribution in the compact and non-compact settings, obtain non-asymptotic estimates using the Cwikel-Lieb-Rozenblum inequality, and estimate the trace of the associated heat-kernel assuming that the associated heat semi-group is hypercontractive. As a side note, an interesting trichotomy for the heat-kernel is obtained. so that the usual integration by parts formula is satisfied with respect to µ:Here ∇ g denotes the Levi-Civita connection, ∆ g denotes the usual Laplace-Beltrami operator, and we use ·, · = g. One immediately sees that −∆ g,µ is a symmetric and positive semi-definite linear operator on L 2 (µ) with dense domain C ∞ c (M ), the space of compactly supported smooth functions on M . In fact, it is well-known (e.g. [8, Proposition 3.2.1]) that the completeness of (M, g) ensures that −∆ g,µ is essentially self-adjoint on the latter domain, and so its graph-closure is its unique self-adjoint extension. We continue to denote the resulting positive semi-definite self-adjoint operator by −∆ g,µ , with corresponding domain Dom(∆ g,µ ). By the spectral theory of self-adjoint operators (see Subsection 3.2), the spectrum σ(−∆ g,µ ) is a subset of [0, ∞). When the spectrum is discrete (such as for compact manifolds), it is composed of isolated eigenvalues of finite multiplicity which increase to infinity; we denote these by λ k = λ k (M n , g, µ), and arrange them in non-decreasing order (repeated by multiplicity) 0 ≤ λ 1 ≤ λ 2 ≤ . . .. In the discrete case, when M is connected and µ has finite mass, we always have 0 = λ 1 < λ 2 . For the standard definition of {λ k } when the spectrum is possibly non-discrete (as the first eigenvalues until the bottom of the essential spectrum), we refer to Subsection 3.2. In this work, we would like to investigate the spectrum of various classes of weighted-manifolds.Definition. The weighted-manifold (M n , g, µ) satisfies the Curvature-Dimension condition CD(ρ, N ), ρ ∈ R and N = ∞, if: Ric g,µ := Ric g + ∇ 2 g V ≥ ρ g (1.1)as...

show abstract

“…Spectral estimates of elliptic operators represent an important part of the modern spectral theory (see, for example, [2,3,7,9,12,13,14,28]). The classical upper estimate for the first non-trivial Neumann eigenvalue of the Laplace operator (the was proved by Szegö [31] for simply connected planar domains via a conformal mappings technique ("the method of conformal normalization").…”

Section: Introductionmentioning

confidence: 99%

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

Gol’dshtein

Ukhlov

2018

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator

Abstract: The purpose of this manuscript is to present a series of lecture notes on isoperimetric inequalities for the Laplacian, for the Schrödinger operator, and related problems.

Cited by 25 publications

References 84 publications

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

Spectral estimates, contractions and hypercontractivity

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

Contact Info

Product

Resources

About