Spectral Radius Formula for a Parametric Family of Functional Operators

The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. For example, it turns out that the spectral radius of the operator
$$ L_2(\mathbb R^n)\ni u(x)\mapsto u(p^{-1}x+h)-u(p^{-1}x-h)\in L_2(\mathbb R^n),\quad p>1,\quad h\in\mathbb R^n, $$
is equal to $2p^{n/2}$ for transcendental values of $p$, and depends on the coefficients of the minimal polynomial for $p$ in the case where $p$ is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for $p$. The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of $p$.