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A Canonical Gauge for Computing of Eigenpairs of the Magnetic Schrödinger Operator

Jeffrey S. Ovall, Li Zhu

TL;DR

This work addresses efficient and accurate computation of eigenpairs for the magnetic Schrödinger operator $H(oldsymbol{A},V)$ by exploiting gauge invariance. It introduces a canonical gauge $oldsymbol{F}$, computed via a Neumann Poisson problem that imposes $oldsymbol{A}= abla a+oldsymbol{F}$ with $ ablaoldsymbol{F}=0$ and $oldsymbol{F}oldsymbol{n}=0$, ensuring spectral equivalence with $H(oldsymbol{A},V)$ while yielding smoother eigenvectors. The authors provide theoretical results and extensive numerical experiments showing that solving eigenproblems with $H(oldsymbol{F},V)$ allows coarser discretizations and faster convergence without sacrificing accuracy, across multiple geometries and potentials. The practical impact is a robust, cost-effective approach for simulating quantum systems in electromagnetic fields, enabling more scalable and stable eigenvalue computations in applications ranging from physics to engineering.

Abstract

We consider the eigenvalue problem for the magnetic Schrödinger operator and take advantage of a property called gauge invariance to transform the given problem into an equivalent problem that is more amenable to numerical approximation. More specifically, we propose a canonical magnetic gauge that can be computed by solving a Poisson problem, that yields a new operator having the same spectrum but eigenvectors that are less oscillatory. Extensive numerical tests demonstrate that accurate computation of eigenpairs can be done more efficiently and stably with the canonical magnetic gauge.

A Canonical Gauge for Computing of Eigenpairs of the Magnetic Schrödinger Operator

TL;DR

This work addresses efficient and accurate computation of eigenpairs for the magnetic Schrödinger operator by exploiting gauge invariance. It introduces a canonical gauge , computed via a Neumann Poisson problem that imposes with and , ensuring spectral equivalence with while yielding smoother eigenvectors. The authors provide theoretical results and extensive numerical experiments showing that solving eigenproblems with allows coarser discretizations and faster convergence without sacrificing accuracy, across multiple geometries and potentials. The practical impact is a robust, cost-effective approach for simulating quantum systems in electromagnetic fields, enabling more scalable and stable eigenvalue computations in applications ranging from physics to engineering.

Abstract

We consider the eigenvalue problem for the magnetic Schrödinger operator and take advantage of a property called gauge invariance to transform the given problem into an equivalent problem that is more amenable to numerical approximation. More specifically, we propose a canonical magnetic gauge that can be computed by solving a Poisson problem, that yields a new operator having the same spectrum but eigenvectors that are less oscillatory. Extensive numerical tests demonstrate that accurate computation of eigenpairs can be done more efficiently and stably with the canonical magnetic gauge.
Paper Structure (13 sections, 2 theorems, 23 equations, 11 figures, 8 tables)

This paper contains 13 sections, 2 theorems, 23 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Gauge Invariance and a Practical Helmholtz Decomposition
  3. Constant vector fields
  4. Constant-curl vector field comparisons
  5. A First Look at Example 2
  6. Numerical Results
  7. Example 1
  8. Example 2
  9. Example 2a: Eigenpairs for $H(\mathbf{A})$, $H(\mathbf{F})$.
  10. Example 2b: Eigenpairs for $H(\mathbf{A},V)$ and $H(\mathbf{F},V)$.
  11. Example 3
  12. Example 4
  13. Conclusion

Key Result

Lemma 2.1

Suppose that $\mathbf{A}=\nabla a + \mathbf{F}$ in $\Omega$for some scalar field $a$ and vector field $\mathbf{F}$. Then $e^{-\mathfrak i a}H(\mathbf{A}, V)e^{\mathfrak i a}=H(\mathbf{F}, V)$. Furthermore, $(\lambda, \psi)$ is an eigenpair of $H(\mathbf{A}, V)$ if and only if $(\lambda, e^{-\mathfra

Figures (11)

  • Figure 1: The first eigenvector $\psi_1$ of $H(\mathbf{A})$ (top row), and the first eigenvector $\phi_1$ of $H(\mathbf{F})$, for Example 2. From left to right in the top row, we see $|\psi_1|$, $\Re\psi_1$, $\Im\psi_1$ and the phase of $\psi_1$; the bottom row is analogous for $\phi_1$.
  • Figure 2: From left to right in the top row, we see $|\mathbf{A}|$, $|\psi_1|$,$|\mathbf{A}\psi_1|$, and $|\nabla \psi_1|$; the bottom row is analogous for $\mathbf{F}$ and $\phi_1$.
  • Figure 3: Streamplot of $\mathbf{A}$ overlaid on plot of $\|\mathbf{A}\|$ (top) and plot of $|\hbox{curl}\mathbf{A}|$ (bottom) for the vector fields used in the experiments.
  • Figure 4: $\Re(\psi_j)$ for $H(\mathbf{A})$ (top row) and $\Re(\phi_j)$ for $H(\mathbf{F})$ (bottom row) when $h=0.01$, $1\leq j\leq 6$, for Example 1.
  • Figure 5: The computed eigenvectors $|\psi_j|$ of $H(\mathbf{A})$, $1\leq j\leq 6$, when $h=0.01$ (top row), $h=0.03$ (middle row), and $h=0.05$, for Example 1.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Lemma 2.1: Gauge Invariance
  • proof
  • Theorem 2.2: Canonical Gauge
  • proof
  • Remark 2.3