On a sum of squares operator related to the Schrödinger equation with a magnetic field (Q6617998)

In this paper, the authors are interested in the analytic and Gevrey regularity properties of the solution of a second-order partial differential equation of the form \N\[\ND_1^2+(D_2+A(x)D_y)^2=f(x,y),\N\]\Nwhere \(A\) is a polynomial in \(x=(x_1,x_2)\in\mathbb{R}^2\), \(y\in\mathbb{R}\), \(D_j=-i\partial_{x_j}\) for \(j=1,2\), and where \(f(x,y)\) is a real analytic function in a neighborhood of the origin of \(\mathbb{R}^3\). Such equation is related to the Schrödinger equation with a magnetic field and no electric field.\N\NUnder the assumptions that the (magnetic) vector potential has some degree of homogeneity and that the Hörmander bracket condition is satisfied, the authors prove in particular that the local analytic/Gevrey regularity of the solution is related to the multiplicities of the zeroes of the Lie bracket of the vector fields.