The scaling limit of eigenfunctions for 1d random Schrödinger operator

Nakano, Fumihiko

doi:10.48550/arxiv.1912.01436

Cited by 1 publication

(2 citation statements)

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“…Actually the point process of eigenvalues of the matrix in the bulk converges to the Sch random point process [KVV12] and the eigenfunctions are delocalized [RV18]. There are also connections with recent investigations in [Nak14, KN17,Nak19] of the aforementioned Russian model of Goldsheid, Molchanov and Pastur [GMP77], in which, as in the random matrix model, a parameter depending on the size of the system is added in front of the potential to reduce its influence.…”

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confidence: 66%

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Localization of the continuous Anderson Hamiltonian in 1-D

Dumaz

Labbé

2019

Probab. Theory Relat. Fields

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We study the bottom of the spectrum of the Anderson Hamiltonian HL := −∂ 2 x + ξ on [0, L] driven by a white noise ξ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L → ∞, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity e x dx, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.

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confidence: 66%

“…It is conjectured in [RV18] that this shape should appear for various critical operators thus its denomination "universal". It was also proved to arise in another random Schrödinger model recently, see [Nak19].…”

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confidence: 93%