Fundamental solution of a Schrödinger parabolic equation with a small parameter. I (Q910553)

It is considered the fundamental solution u for the parabolic equation of Schrödinger type: \[ \partial u/\partial t=(1/2)\Delta_ Mu- (1/\epsilon)q(x)u;\quad u(0,x,y)=\delta_ y(x) \] on the n-dimensional manifold M, where \(\epsilon\) is a small parameter. The main goal of the paper is to evaluate the solution u when \(t\to 0\), \(t^ 2/\epsilon \to 0\), and also when \(t\to 0\), \(t^ 2/\epsilon \to \infty\).