Regularity properties of a double characteristics differential operator with complex lower order terms

On the spectrum of an anharmonic oscillator

“…On the other hand, if α(x, y) is not constant, condition (5.2) is no longer necessary to guarantee the hypoellipticity of G, as we shall show in the sequel (see also [28], [29]. For instance, assume that condition (5.2) is violated at the origin for some j 0 ∈ N; namely, The main result of this section is the following Theorem 5.1.…”

“…and prove the following upper bound for the number N − (H) of the negative eigenvalues of H (see (29) in [8]):…”

Minimal Gevrey Regularity for Hörmander Operators

“…We point out that when the vector field 𝑋 0 is complex the problem is much more involved. We refer to [16], [20], [3] for papers devoted to that case. If 𝑋, 𝑌 , are two vector fields we write [𝑋, 𝑌 ], the commutator, or Lie bracket, of 𝑋 and 𝑌 as Let 𝑟 be a positive integer and let 𝑖 1 , .…”

Analytic Hypoellipticity for Sums of Squares in the Presence of Symplectic Non Treves Strata

“…are just harmless perturbations if x 2 is close to zero and x 3 is bounded, so that the operator is very similar to a sum of anharmonic oscillators in two different variables [13,14]. It is then well known and not too difficult to show that P is microhypoanalytic at (0, e 3 ), i.e., that (0, e 3 ) ∈ W F a (u) if (0, e 3 ) ∈ W F a (Pu), where W F a denotes the analytic wave front set as defined in e.g., [10,Definition 8.4.3].…”