Physics:Linear Schrödinger Operator

A linear Schrödinger operator is an exponential operator used in the theory of Schrödinger equation in expressing solution of Schrödinger equation in an integral form. It is essentially defined by [math]\displaystyle{ U(t) =e^{i tD } }[/math], where [math]\displaystyle{ D=\sum_{i,j}a_{ij}\partial_{x_j}\partial_{x_k} }[/math] in its general form, [math]\displaystyle{ \sqrt{-1}=i }[/math] and [math]\displaystyle{ x \in \mathbb{R}^d }[/math].