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 $R_A(i\hat{I}»)=(A-i\hat{I}»{)}^{-1}$ 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 $\hat{I}“(1/4)$ in two dimensions and Watson's integral in three. For the Laplacian $A=-\hat{I}”$ on ${\mathbb{R}}^n$ we expand $R_A(i\hat{I}»)\ddot{I}†$, for $\ddot{I}†\in L^1({\mathbb{R}}^n)\cap L^2({\mathbb{R}}^n)$, as a series of convolutions with the free Schrödinger 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 $\ddot{I}ƒ(A)$, including, through the limiting absorption principle, the absolutely continuous spectral density of $-\hat{I}”+V$, together with the density of states and the spectral zeta function, the last reducing to the Riemann zeta function for $-\hat{I}”$ on the circle and yielding the functional determinant of $-\hat{I}”+m^2$ 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.