Fundamental solution of a Schrödinger-type parabolic equation with small parameter. II (Q919174)

[For part I see ibid. 44, No.4, 1-6 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No.4, 3-6 (1989; Zbl 0696.35014).] Let \(I(\phi)=(1/2)\int^{1}_{0}| {\dot \phi}(\tau)|^ 2 d\tau +k\int^{1}_{0}q(\phi (\tau))d\tau\), \(\phi (0)=x\), \(\phi (1)=y\) and \(V(x,y)=\inf I(\phi)\). The author proves that the problem \[ \partial u/\partial t=(1/2)\Delta u-(1/\epsilon)q(x)u,\quad u(0,x,y)=\delta_ y(x), \] has a solution \(u_{\epsilon}(t,x,y)\sim_{t\to 0}(Q(x,y)/(2\pi t)^{\nu /2})\exp [V(x,y)/t-kC(x,y)]\).