Quantum unique ergodicity for random bases of spectral projections

Maples, Kenneth

doi:10.4310/mrl.2013.v20.n6.a10

Cited by 16 publications

(20 citation statements)

References 10 publications

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“…In [2], the quantum ergodic property was generalized to any compact Riemannian manifold, with H N the span of the eigenfunctions in a spectral interval [N, N + 1] for √ ; in [3], essentially the same result was proved for holomorphic sections of line bundles over Kähler manifolds. Related results for eigenfunctions have recently been proved in [4,5]. The dimension of such H N grows at the rate N m−1 , where m = dim M, and thus a random element of H N is a superposition of N m−1 states.…”

Section: Maximal Torus T D N ⊂ U(d Nmentioning

confidence: 98%

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Quantum ergodicity of random orthonormal bases of spaces of high dimension

Zelditch

2014

Phil. Trans. R. Soc. A.

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We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S 2 . In particular, it holds on the flat torus if d ≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.

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Section: Maximal Torus T D N ⊂ U(d Nmentioning

confidence: 98%

“…Recently, this has been proved in [4,5] when the spectral intervals satisfy certain growth assumptions. …”

Section: Maximal Torus T D N ⊂ U(d Nmentioning

confidence: 99%

Quantum ergodicity of random orthonormal bases of spaces of high dimension

Zelditch

2014

Phil. Trans. R. Soc. A.

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“…-Our proof of unique quantum ergodicity for random bases can be adapted to prove analogous results for the Laplace-Beltrami operator on compact manifolds. Notice that our method is different from the method used in [28,14]. In particular we do not use the Szegö limit theorem like in [28, prop.…”

Section: (Per)mentioning

confidence: 99%

Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

Robert

Thomann

2015

Commun. Math. Phys.

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This paper is a continuation of Poiret et al. (Ann Henri Poincaré 16:651-689, 2015), where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential V on R d . We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on R d . This construction relies on accurate estimates on the spectral function in a noncompact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all bases of Hermite functions are quantum uniquely ergodic.

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“…A similar construction that involves partitioning the spectrum of the Laplacian appropriately can be used to make sense of random bases (defined using either Haar measures or Wigner induced measures on unitary groups) on any compact Riemannian manifold. Readers are referred to [Z3,Z4,M,BL] for the general construction. A natural next step is to extend our quantum ergodicity result to Wigner induced random bases of Laplacian eigenfunctions or approximate eigenfunctions on other manifolds.…”

Section: Introductionmentioning

confidence: 99%

Quantum ergodicity of Wigner induced random spherical harmonics

Chang

2018

J. Spectr. Theory

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We introduce a new notion of a 'random orthonormal basis of spherical harmonics' of L 2 (S 2 ) using generalized Wigner ensembles and show that such a random basis is almost surely quantum ergodic. Similar quantum ergodicity results (with varying degrees of generality) are obtained in [Z2, Z3, Z4, M, BL] for random Laplacian eigenfunctions defined using Haar measures on unitary groups. Our main contribution comes from the use of a more general measure than previously studied, as the Gaussian unitary ensemble (which induces Haar measure on the unitary group) is a special case of the generalized Wigner ensemble. We are able to work with this more general class of measures because Wigner eigenvectors are asymptotically Gaussian, a result proved in [KY, TV] (with additional assumptions on the moments) and [BY]. Our quantum ergodicity statement also provides a semi-classical realization of the probabilistic 'local quantum unique ergodicity' of [BY].

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Quantum unique ergodicity for random bases of spectral projections

Cited by 16 publications

References 10 publications

Quantum ergodicity of random orthonormal bases of spaces of high dimension

Quantum ergodicity of random orthonormal bases of spaces of high dimension

Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

Quantum ergodicity of Wigner induced random spherical harmonics

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