On a ``Hamiltonian path-integral'' derivation of the Schrödinger equation
This paper deals with the Cauchy problem for the Schrödinger equation with an external electro-magnetic potential on \(\mathbb{R}^m_x\). The author proves that there exists a family of unitary operators \({\mathcal U}(t,s)\), \((t,s)\in [-T,T]^2\), \(T>0\), \(T\) fixed on \(L^2(\mathbb{R}^m)\) and such that \(U(s,s)= \text{Id}\), \(U(t_1,t_2)U(t_2,t_3) u= U(t_1,t_3)u\), \(\forall u\in L^2(\mathbb{R}^m)\) and for any \(t_1,t_2,t_3\in [-T, T]\). Moreover, for each \(u\in{\mathcal S}(\mathbb{R}^m)\), \(U(t,s)u\) satisfies the Schrödinger equation. According to the author: ``We construct a parametrix which exhibits clearly how quantities from the Hamiltonian (not Lagrangian) mechanics are related to quantum mechanics.'' The parametrix \(U(t,s)\) is approximated by a Cauchy net of Fourier integral operators on \(\mathbb{R}^m\).
scientific article; zbMATH DE number 1455441