Method of reflections applied to the construction of the asymptotics of Green's function for parabolic equations with variable coefficients (Q1907435)

A method for constructing the asymptotic representation of Green's function for the equation \[ -h {\partial G\over \partial t} +H\left( {\overset {2} x}, -h{\overset {1} {{\partial \over \partial x}}} \right) G=0, \quad x\geq 0 \tag{1} \] as \(h\to 0\) is presented. Here, the Hamiltonian \(H(x,p)\) of equation (1) is an analytic function in \(p\), infinitely differentiable in \(x\) and real-valued for real values of the arguments, and satisfies for \((x,p,\eta) \in\mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}\) the conditions \(\partial^2 H(x,p)/\partial p^2>0\), \(\text{Re} H(x,p+i \eta) \leq H(x,p)\). The symbols \({\overset {2} x}\), \(\overset {1}{{\partial \over \partial x}}\), are as introduced by \textit{V. P. Maslov} [Operational methods, Moscow, Mir (1976; Zbl 0449.47002)]. The method is based on a particular representation of the \(\delta\)-function obtained by the first author.