Essential Self–adjointness and Global Hypoellipticity of the Twisted Laplacian

A scale of Sobolev spaces is introduced to measure the global hypoellipticity of the twisted Laplacian. The essential self-adjointness of the twisted Laplacian is established and the domain of the unique self-adjoint extension is determined in terms of the Sobolev spaces. The global hypoellipticity of the twisted Laplacian in the Gelfand–Shilov spaces is also proved. 1. The twisted Laplacian Let ∂ ∂z and ∂ ∂z be linear partial differential operators on R 2 given by ∂ ∂z = ∂ ∂x − i ∂ ∂y and ∂ ∂z = ∂ ∂x + i ∂ ∂y . Then we define the linear partial differential operator L on R2 by L = − 2 (ZZ + ZZ), where Z = ∂ ∂z + 1 2 z, z = x− iy, and Z = ∂ ∂z − 1 2 z, z = x + iy. The vector fields Z and Z, and the identity operator I form a basis for a Lie algebra in which the Lie bracket of two elements is their commutator. In fact, −Z is the formal adjoint of Z and L is an elliptic partial differential operator on R2 given by L = −∆ + 1 4 (x2 + y2)− i (