Higher energy state approximations in the “Many interacting worlds” model

In the “Many Interacting Worlds” (MIW) discrete Hamiltonian system approximation of Schrödinger’s wave equation, introduced in Ref. 11, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein’s method, in Wasserstein-$1$ distance with rate ${𝒪}(\sqrt{logN}/N)$ has been shown Refs. 5, 13, and 15. In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-1 distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the “differential equation” approach to Stein’s method, which allows to handle behavior near zeros of the higher energy states.