On the Green function of a slightly heterogeneous waveguide

The uniform asymptotic expansion on powers of a small parameter $\varepsilon$ for a solution of the problem is $[\Delta+k^2(\varepsilon x, y)]u(x, y)=\delta(x-x_0)\delta(y-y_0)$, $u(x, 0)=u(x, H)=0$ obtained.