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Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on $\mathbb{R}^2$

Xuefeng Liu

Abstract

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator $H=-Δ+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the restricted eigenvalue problem is solved with a rigrous eigenvalue bound, where Liu's eigenvalue bound along with the Composite Enriched Crouzeix--Raviart (CECR) finite element method proposed plays a central role. Two concrete potentials are studied: the radially symmetric ring potential $V_1(x)=(|x|^2-1)^2$ and the Cartesian double-well $V_2(x)=(x_1^2-1)^2+x_2^2$. To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain.

Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on $\mathbb{R}^2$

Abstract

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator with a confining potential on . The method combines domain truncation to a finite disk on which the restricted eigenvalue problem is solved with a rigrous eigenvalue bound, where Liu's eigenvalue bound along with the Composite Enriched Crouzeix--Raviart (CECR) finite element method proposed plays a central role. Two concrete potentials are studied: the radially symmetric ring potential and the Cartesian double-well . To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain.

Paper Structure

This paper contains 19 sections, 4 theorems, 38 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. General Framework
  4. Two Model Potentials
  5. $V_1(x)=(|x|^2-a^2)^2$.
  6. $V_2(x)=(x_1^2-a^2)^2+x_2^2$.
  7. Methodology
  8. Dirichlet Upper Bounds
  9. Lagrange $P_1$ upper bounds: $\lambda_{k,h}^{R,\mathrm{P1}}$.
  10. Neumann Exterior Lower Bound
  11. CECR FEM and Lower Bound
  12. ECR element.
  13. CECR composite space.
  14. Interpolation error and $C_h$.
  15. Liu's Lower Bound Theorem
  16. ...and 4 more sections

Key Result

Proposition 1

$\lambda_k\le\lambda_k^R$ for all $k\ge1$.

Theorems & Definitions (12)

  • Proposition 1: Upper bound
  • proof
  • Remark 1: Lower bound using Lagrange $P_1$ FEM
  • Lemma 1: Restriction lower bound
  • proof
  • Proposition 2: Exterior lower bound
  • proof
  • Remark 2
  • Theorem 1: Liu liu2015
  • Remark 3: Agmon decay and Neumann approximation
  • ...and 2 more