# Spectral statistics for random Schrödinger operators in the localized regime

François Germinet; Frédéric Klopp

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 9, page 1967-2031
- ISSN: 1435-9855

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topGerminet, François, and Klopp, Frédéric. "Spectral statistics for random Schrödinger operators in the localized regime." Journal of the European Mathematical Society 016.9 (2014): 1967-2031. <http://eudml.org/doc/277236>.

@article{Germinet2014,

abstract = {We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda $. Consider now $I_\{\Lambda \}$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda $ grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive • uniform Poisson behavior of the locally unfolded eigenvalues, • a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales, • the distribution of the unfolded level spacings, locally and globally, • the distribution of the unfolded localization centers, locally and globally.},

author = {Germinet, François, Klopp, Frédéric},

journal = {Journal of the European Mathematical Society},

keywords = {random Schrödinger operators; eigenvalue statistics; level spacing distribution; random Schrödinger operators; eigenvalue statistics; level spacing distribution},

language = {eng},

number = {9},

pages = {1967-2031},

publisher = {European Mathematical Society Publishing House},

title = {Spectral statistics for random Schrödinger operators in the localized regime},

url = {http://eudml.org/doc/277236},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Germinet, François

AU - Klopp, Frédéric

TI - Spectral statistics for random Schrödinger operators in the localized regime

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 9

SP - 1967

EP - 2031

AB - We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda $. Consider now $I_{\Lambda }$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda $ grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive • uniform Poisson behavior of the locally unfolded eigenvalues, • a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales, • the distribution of the unfolded level spacings, locally and globally, • the distribution of the unfolded localization centers, locally and globally.

LA - eng

KW - random Schrödinger operators; eigenvalue statistics; level spacing distribution; random Schrödinger operators; eigenvalue statistics; level spacing distribution

UR - http://eudml.org/doc/277236

ER -

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