When is the Fourier transform of an elementary function elementary?

By an elemementary function on an n-dimensional complex vector space $V$ we will mean a complex-valued function of the form $f(x,h)=A(x)*\exp(i*\mathrm{Re}Q(x)/h)$, where $Q(x)$ is a rational function, h is a positive parameter, and $A(x)$ is a product of (complex) powers of polynomials of $x$ and powers their complex conjugates. Such a function can be considered as a distribution on $C^n$. We will study the following question: when is the Fourier transform of $f(x,h)$ also elementary for all values of $h$? The simplest example of this is the Gaussian $f(x)=\exp(i*\mathrm{Re}Q(x)/h)$, where $Q$ is a nondegenerate quadratic form, and we will be interested in studying other possible examples. First, by sending $h$ to zero and using the stationary phase method, one can derive a necessary condition of this (“the semiclassical condition”): the differential $dQ$ is a birational isomorphism between $V$ and $V^*$ (i.e. the inverse map to $dQ$ is rational). This is equivalent to saying that the Legendre transform of $Q$ is rational. It is clear that a homogeneous function $Q$ satisfying the semiclassical condition must be of degree 0 or 2; but this is certainly not sufficient. In fact, the problem of classification of homogeneous functions $Q$ satisfying the semiclassical condition is very interesting. For example, assume that $V=W+C$ (so $x=(y,t)$) and $Q(x)=f(y)/t$, where $f$ is an irreducible cubic polynomial on $W$. Then $W$ is the complexified space of 3 by 3 hermitian matrices over division rings $R$, $C$, $H$ or $O$ (so $\dim W$ is 6, 9, 15, or 27), and f is proportional to the the determinant polynomial. This is proved using a beautiful theorem of F. L. Zak on the classification of Severi varieties. On the other hand, one can show that any (Laurent) monomial in variables $x_1,\dots,x_n$ of degree 0 or 2 satisfies the semiclassical condition. So one may wonder which of such monomials $Q$ give rise to elementary functions $f$ with elementary Fourier transform, (assuming that $A$ is a product of powers of coordinates and powers of conjugare coordinates). In other words, which $Q$ satisfy the “quantum condition”?. The answer turns out to be unexpectedly interesting. For example, suppose that $Q=x_n^m/y_1^{m_1}\dots y_{n-1}^{m_{n-1}}$, where $m=m_1+\dots+m_{n-1}+2$. Then functions $f$ (i.e. choices of $A$) for which the Fourier transform of f is elementary are (up to scaling) in bijection with exact covering systems of type $(m_1,\dots,m_{n-1},1,1)$; we recall that an exact covering system of type $(p_1,\dots,p_k)$ is a covering of the group $Z/pZ$, where $p=p_1+\dots+p_k$, by cosets of $Z/p_kZ$, one copy of each (so $p_k$ should divide $p$ for such a system to exist). Thus, the monomial $x^3/y$ satisfies the quantum condition, while the monomial $x^5/y^3$ does not.