Momentum operators in the unit square (Q395615)

The authors extend previous work on the self-adjoint extensions of the momentum operator \(P_{\min}:=\frac{1}{2\pi\,i}\frac{d}{dx}\) acting on a suitable space to a two dimensional situation. More specifically, they consider the operator \(\frac{1}{2\pi\,i}\frac{\partial}{\partial x}\) defined on a domain \(C_c^\infty([0,1]^2)\), where it is symmetric, and look for its self-adjoint extensions. These extensions are in one-to-one correspondence with some unitary operators \(V\) acting on \({\mathcal L}^2([0,1])\). After this preliminary analysis, the authors apply their procedure to investigate the characterization of commuting self-adjoint extensions of \(\frac{1}{2\pi\,i}\frac{\partial}{\partial x}\) and \(\frac{1}{2\pi\,i}\frac{\partial}{\partial y}\) acting on different sets. In particular, they consider the square \([0,1]^2\), the strip \([0,1]\times{\mathbb R}\) and the set \([0,1]\times C\), where \(C\) is some fractal set.