 
Summary: ANNALS OF PHYSICS 143, 3353 (1982)
The Lifetime of Wannier Ladder States*
JOSEPH E. AVRON
Lauritsen Laboratory, Calfornia Institute of Technolog.v,
Pasadena, California 91125
Received January 25, 1982
I study the width of the Wannier ladder states, i.e, Bloch electrons in external homogeneous
field. For periodic potentials with a finite number of gaps, a formula for the width is obtained
showing that the width vanishes exponentially fast with the field in accordance with Zener
tunneling. The case of infinitely many gaps is studied qualitatively, and it is argued that
although the width decreases exponentially "on the average," the detailed bahavior is very
complicated. In particular the width oscillates over different orders of magnitude as the field
changes slightly. The oscillations are a consequence of a resonance phenomenon.
1. THE PROBLEM
The Wannier ladder (also referred to as "Stark ladder" or "WannierStark
ladder") (26,271 is a ladder of resonances for the one dimensional Bloch electron in
a homogeneous electric' field f :*
Hf = p* + V(x)  fx, V(x + 2x) = V(x), s >o,
. d (1.1)
p=tdx*
