MIW Sequences

Abstract

The Many Interacting Worlds (MIW) discrete Hamiltonian system approximation of the Schrödinger wave equation was introduced in [10]. Building on this work, convergence of MIW ground states to the normal ground state of the quantum harmonic oscillator has been shown via Stein's method [13, 5,15, 12]. In the Wasserstein-1 metric, this occurs with rate $\calO(\sqrt{\log N}/N)$. In this context, we define a general notion of MIW sequences as a kind of discrete approximation of continuous functions. MIW sequences are defined by the relation\[ \frac{1}{x_{n+1} - x_n} - \frac{1}{x_n - x_{n-1}} = \frac{f'(x_n)}{f(x_n)}, \] along with boundary conditions at $\pm\infty$. This is an approximation in the sense that given a function $f$, the empirical distributions of the corresponding MIW sequences should converge to the measure induced by $f$, i.e.\ $f\ dx$, as the number of sequence points diverges. We show the existence of MIW sequences derived from $C^2$, positive, integrable, locally log-concave functionswith finitely many zeros. We also show convergence in Wasserstein-1 distance of the empirical distributions of MIW sequences derived from positive, integrable functions of the form $p(x) e^{-q(x)}$, where both $p$ and $q$ are non-zero polynomials. As well, we show that MIW sequences derived from solutions to the time-independent Schrödinger equation are stationary, pointwise in the limit, under the corresponding MIW mechanics. The classes of functions considered all include the PDF of the normal distribution, as well as the other time-independent stable states of the quantum harmonic oscillator, so these results extend the work published in [12].