**Author:**
Guido Gentile

**Title:** *
Quasi-periodic solutions for two-level systems
*

**Abstract: **
We consider the Schroedinger equation for a class of
two-level atoms in a quasi-periodic external field in the case in which
the spacing 2e between the two unperturbed energy levels is small.
We prove the existence of quasi-periodic solutions
for a Cantor set E of values of e around the origin
which is of positive Lebesgue measure:
such solutions can be obtained from the formal power series
by a suitable resummation procedure.
The set E can be characterized by requesting
infinitely many Diophantine conditions of Mel'nikov type.

**Keywords:**
Two-level systems, generalized Riccati equation,
quasi-periodic solutions, trees,
resummation of divergent series, Cantor set

Guido Gentile

Dipartimento di Matematica

Universita` di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it