Nonvanishing of the Cole–Hopf denominator for complex deformation parameter

Determine whether the quantity (e^{(t/R_e) ∂_x^2} e^{-(δ R_e/2) ∫_{-∞}^x g(y) dy})(x) has strictly positive complex modulus for all t > 0, x ∈ R, and complex δ when the initial data g ∈ L^1(R); equivalently, ascertain whether the denominator (e^{t 𝓛_δ e^{-(δ R_e/2) 𝓣g}})(x) in the Cole–Hopf representation of the linear homotopy Burgers’ solution can ever vanish for Im(δ) ≠ 0.

Background

To establish holomorphic dependence on the deformation parameter δ and infinite radius of convergence of the series expansion for the linear homotopy Burgers’ equation, the authors express the solution via the Cole–Hopf transform as a ratio of two quantities involving the semigroup e^{t 𝓛_δ} acting on e^{-(δ R_e/2) 𝓣g}.

They show the denominator is strictly positive for real δ but could not rule out zeros for complex δ with nonzero imaginary part. Proving nonvanishing for all (t,x,δ) would eliminate non-removable singularities in δ and yield global analyticity of u(t,x;δ) without restricting to almost-everywhere in x.