Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?


For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. I'm wondering if there is a known example where the angle is e.g., $\pi/4$ or other value $< \pi/2$.



Angle of analyticity of semigroup
Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \nabla$, where $B$ is a matrix. Assuming that all eigenvalues of $B$ have negative real parts, then an invariant measure $\mu$ exists (and is given by a Gaussian density). It turns out that the angle of analiticity in $L^p$ of the invariant measure can be computed exactly and can be smaller than $\pi/2$, even for $p=2$. This can be found in a paper by Chill, Fasangova, Pallara and myself. 


*Chill, R.; Fašangová, E.; Metafune, G.; Pallara, D.*, [**The sector of analyticity of the Ornstein-Uhlenbeck semigroup on (L^p) spaces with respect to invariant measure**](http://dx.doi.org/10.1112/S0024610705006344), J. Lond. Math. Soc., II. Ser. 71, No. 3, 703-722 (2005). [ZBL1123.35030](https://zbmath.org/?q=an:1123.35030).


To obtain similar examples in unweighted spaces one has only (but patiently) to compensate the weight thus obtaining an operator with a linear drift and a quadratic potential. This works however only from dimension $2$ on; in the one dimensional case the operator is always self-adjont but still the angle of analitycity is different from $\pi/2$ in $L^p$, for $p$ different from $2$. There is also a paper by E. Priola and myself dealing with non-analytic Markov semigroups where one finds other examples. 


*Metafune, G.; Priola, E.*, [**Some classes of non-analytic Markov semigroups**](http://dx.doi.org/10.1016/j.jmaa.2004.02.037), J. Math. Anal. Appl. 294, No. 2, 596-613 (2004). [ZBL1067.47055](https://zbmath.org/?q=an:1067.47055).


It would be nice to have a direct approach, avoiding the detour.


