We prove that the eigenvalues λn​(c) of the time-frequency
localization operator satisfy λn​(c)>1−δc for n=[(1−ε)c], where δ=δ(ε)<1 and ε>0 is arbitrary, improving on the result of Bonami, Jaming and Karoui, who
proved it for ε≥0.42. The proof is based on the properties of
the Bargmann transform.Comment: 13 page