**Simona Fornaro**

*Department of Mathematics, University of Pavia*

In this talk I will describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators A in $L^p$ spaces with respect to the Lebesgue measure, for any $1 \le p < +\infty$. Several applications in physics and finance for A and its evolutionary counterpart A- $D_t$ can be found in the literature. Moreover these operators have been the leading example for an intensive research on elliptic and parabolic problems with unbounded coefficients.

I will show that the spectrum of A is the sum of the spectrum of the diffusion part and the spectrum of the drift term. This result completes the picture of the spectral properties of Ornstein-Uhlenbeck operators in $L^p$ spaces.