Generalized vector measures and path integrals for hyperbolic systems (Q915988)

The author considers the Cauchy problem given by \[ (\partial /\partial t)\Psi (t,x)-[\sum^{d}_{t=1}P_ t(\partial /\partial x_{\ell})+iQ+V(x)]\Psi (t,x)=0,\quad 0<t<T,\quad x\in {\mathbb{R}}^ d,\quad \Psi (0,x)=g(x), \] where \(0<T\leq \infty\), and V(x) is a complex- valued bounded Borel measurable function while \(P_{\ell}\) (1\(\leq \ell \leq d)\) and Q are constant Hermitian \(N\times N\) matrices. He supposes that the matrices \(P_{\ell}\) are not simultaneously diagonalizable, which makes the problem not \(L^{\infty}\) well-posed but only \(L^ 2\) well-posed. His aim is to establish a path integral formula for the solution of this problem, namely: \[ \Psi (t,)=\int_{\tilde x_ t}d\mu_ t(x) \exp \{\int^{t}_{0}V(x(s))ds\} g(x(0)). \] Here X: [0,t]\(\to {\mathbb{R}}^ d\) is a Lipschitz continuous path and \(\tilde x_ t\) is the space of these functions while \(\mu_ t\) stands for an \({\mathcal L}(L^ 2({\mathbb{R}}^ d;{\mathbb{C}}^ N))\)-valued generalized vector measure on \(\tilde x_ t\). The main results of the paper show the existence of the measure \(\mu_ t\).