42 research outputs found
A Lower Bound on the Ground State Energy of Dilute Bose Gas
Consider an N-Boson system interacting via a two-body repulsive short-range
potential in a three dimensional box of side length . We take
the limit while keeping the density fixed
and small. We prove a new lower bound for its ground state energy per particle
as , where is the scattering length of .Comment: 26 pages, AMS LaTe
Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices
We consider random matrices of the form ,
, where is a real symmetric or complex Hermitian
Wigner matrix of size and is a real bounded diagonal random matrix of
size with i.i.d.\ entries that are independent of . We assume
subexponential decay for the matrix entries of and we choose , so that the eigenvalues of and are typically of the same
order. Further, we assume that the density of the entries of is supported
on a single interval and is convex near the edges of its support. In this paper
we prove that there is such that the largest
eigenvalues of are in the limit of large determined by the order
statistics of for . In particular, the largest
eigenvalue of has a Weibull distribution in the limit if
. Moreover, for sufficiently large, we show that the
eigenvectors associated to the largest eigenvalues are partially localized for
, while they are completely delocalized for
. Similar results hold for the lowest eigenvalues.Comment: 47 page
Edge Universality for Deformed Wigner Matrices
We consider random matrices of the form where is
a real symmetric Wigner matrix and a random or deterministic, real,
diagonal matrix whose entries are independent of . We assume subexponential
decay for the matrix entries of and we choose so that the eigenvalues
of and are typically of the same order. For a large class of diagonal
matrices we show that the rescaled distribution of the extremal eigenvalues
is given by the Tracy-Widom distribution in the limit of large . Our
proofs also apply to the complex Hermitian setting, i.e., when is a complex
Hermitian Wigner matrix