Dyadic Resolvent Representations of Self-Adjoint Operators: Propagator Expansions, Spectral Measures, and Zeta Functions

Abstract

For a self-adjoint operator $A$ on a Hilbert space, the dyadic resolvent representation of Castillo, Costin and Costin expresses the resolvent $R_A(i\lambda)=(A-i\lambda)^{-1}$ as a series in the unitary group $U_t=e^{-itA}$ sampled at the dyadic times $\{2^{-k}\}_{k\ge 0}$. We develop this representation structurally and through several spectral applications. We first exhibit the underlying operators as a one-parameter scale family obeying a Landen doubling recursion whose telescoping recovers the representation, and record its readings as a series of Zak transforms and as a dyadic filter bank. As a worked instance of the Zak-transform reading we obtain dyadic-Bessel series for the lattice Green functions of $\mathbb{Z}^d$, with closed-form special values: the lemniscatic constant $\Gamma(1/4)$ in two dimensions and Watson's integral in three. For the Laplacian $A=-\Delta$ on $\mathbb{R}^n$ we expand $R_A(i\lambda)\varphi$, for $\varphi \in L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$, as a series of convolutions with the free Schr\"odinger propagator, and derive explicit dyadic representations of the fundamental solutions of the Laplace and Poisson equations in $\mathbb{R}^3$ and of the one-dimensional heat equation. Finally, we reconstruct spectral data of $A$ from the dyadic representation: the spectral measures on $\sigma(A)$, including, through the limiting absorption principle, the absolutely continuous spectral density of $-\Delta+V$, together with the density of states and the spectral zeta function, the last reducing to the Riemann zeta function for $-\Delta$ on the circle and yielding the functional determinant of $-\Delta+m^2$ there. We close by showing that the dyadic samples determine $A$ uniquely, an exact anti-aliasing of the propagator, so that all of this spectral data is a function of the dyadic samples alone.