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Higher energy state approximations in the `Many Interacting Worlds' model

Alex Loomis, Sunder Sethuraman

Abstract

In the `Many Interacting Worlds' (MIW) discrete Hamiltonian system approximation of Schrödinger's wave equation, introduced in \cite{hall_2014}, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein's method, in Wasserstein-$1$ distance with rate $\mathcal{O}(\sqrt{\log N}/N)$ has been shown in McKeague-Levin (2016), Chen-Thanh (2023), McKeague-Swan (2023). 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.

Higher energy state approximations in the `Many Interacting Worlds' model

Abstract

In the `Many Interacting Worlds' (MIW) discrete Hamiltonian system approximation of Schrödinger's wave equation, introduced in \cite{hall_2014}, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein's method, in Wasserstein- distance with rate has been shown in McKeague-Levin (2016), Chen-Thanh (2023), McKeague-Swan (2023). In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein- 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.
Paper Structure (28 sections, 53 theorems, 137 equations, 2 figures)

This paper contains 28 sections, 53 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General aims
  3. Informal statement of results
  4. MIW sequences and higher energy functions
  5. Proof ideas for the main theorems
  6. Plan of the article
  7. Results
  8. Remarks
  9. Proof of Theorem \ref{['thm:exist-unique']}: Existence and uniqueness of MIW sequences
  10. Existence of the functions $\chi_n$
  11. Properties of the functions $\chi_n$
  12. Existence of MIW sequences for higher energy functions when $\ell\geq 1$
  13. Uniqueness of MIW sequences for higher energy functions when $\ell\geq 0$
  14. Proof of Theorem \ref{['thm:gaps']}: Gap sizes and span of MIW sequences
  15. Proof of Theorem \ref{['thm:converge']}: Convergence in Wasserstein-$1$ distance
  16. ...and 13 more sections

Key Result

Theorem 2.1

There is a unique MIW sequence $(x_n)_{n=1}^N$ of $f$ that satisfies the left boundary condition at $x_0=-\infty$, and the right boundary condition at $x_{N+1}=\infty$, with $N_k$ points that lie in the region $R_k$ for each $0 \leq k \leq \ell$, and $N = \sum_{k=0}^{\ell}N_k$.

Figures (2)

  • Figure 1: A simulation of MIW $N=5$ dynamics.
  • Figure 2: MIW $N=5$ dynamics starting from the Maxwellian $\ell=1$ approximating state $(x_n)_{n=1}^5$ with $3$ negative and $2$ positive values.

Theorems & Definitions (103)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 93 more