I'd like to know if does there exist real, continuous and differentiable functions $\omega(x)$ such that
$$ \int_{-\infty}^{\infty} e^{-\frac{x^2}{2} +~ i\omega(x)}dx = 0$$
and what kind of tools one could use to prove their existence and construct them, if possible.
The reason behind this is that I'm trying to take a set of real functions $g_n$ of the type $$g_n(x) = e^{-\frac{(x + nL)^2}{2}}, x, L \in \mathbb{R}, n \in \mathbb{Z}$$
and transform them into complex functions
$$g_n'(x) = g_n(x)e^{i\omega_n(x)}$$
with $\omega_n(x)$ real and such that
$$ A\int_{-\infty}^{\infty} g_n'(x)\overline{g_m'(x)}dx = \delta_{nm}$$
where $A$ is a normalization constant. In Physics, this set of functions, if it exists, would define a discrete orthogonal basis, which can be used to derive the eigenstates and eigenenergies of periodic potentials, such as a crystal. Hence the title.
The closest I got to a solution was finding out that if we define $\omega(x) = \nu x,~\nu \in \mathbb{R}$, then we have that, according to Wolfram Alpha website,
$$ A\int_{-\infty}^{\infty} g_n'(x)\overline{g_m'(x)}dx = A\int_{-\infty}^{\infty} e^{-\frac{(x +nL)^2 + (x+mL)^2}{2}} e^{i(n-m)\nu x}dx = A'e^{-\frac{(n-m)^2\nu^2}{2}}, for ~n\neq m,$$ that is, we can get as small as we want, but never get to zero. I know this just because of the computational tool and I'm curious about how one would prove this last equation.
P.S.: Since I don't come from a mathematical background I'm not even sure how to tag this. Does this belong to functional analysis? Or integral equations, perhaps?