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, J. Lond. Math. Soc., II. Ser. 71, No. 3, 703-722 (2005). ZBL1123.35030.

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, J. Math. Anal. Appl. 294, No. 2, 596-613 (2004). ZBL1067.47055.

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

There are examples but you need singularities or unbounded coefficients; the uniformly parabolic case with regular coefficients is indeed a perturbation of the laplacian. In 1D, if you perturb the harmonic oscillator $$D^2-x^2$$ by a linear drift $$bxD$$, the angle of analyticity depends on $$b$$.