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A multiscale finite element method for the Schrödinger equation with multiscale potentials

Jingrun Chen, Dingjiong Ma, Zhiwen Zhang

TL;DR

The paper tackles solving the Schrödinger equation in the semiclassical regime with multiscale potentials, where traditional asymptotic approaches fail. It introduces a localized multiscale finite element method that builds basis functions via a constrained energy-minimization on a coarse mesh, using sparse compression of the Hamiltonian to achieve $H = O(ε)$ and a time step $k$ independent of $ε$; time evolution is computed through a one-shot eigendecomposition or Jordan-form based solver on the reduced space. The approach yields exponential decay of the basis, enabling localization and a sparse system while preserving gauge invariance and conserving mass and energy. Numerical tests in1D and 2D across various multiscale potentials confirm first-order convergence in $H^1$ and second-order convergence in $L^2$, demonstrating robustness and efficiency for complex quantum heterostructures and metamaterials. The method offers a practical framework that avoids explicit additive-scale decompositions and suggests potential extensions to stochastic inputs and spintronic device modeling.

Abstract

In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptoics-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis are constructed using sparse compression of the Hamiltonian operator, and thus are "blind" to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is $ H=\mathcal{O}(ε) $ where $ε$ is the semiclassical parameter and the time stepsize $ k$ is independent of $ε$. Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in $H^1$ and $L^2$ norms, respectively.

A multiscale finite element method for the Schrödinger equation with multiscale potentials

TL;DR

The paper tackles solving the Schrödinger equation in the semiclassical regime with multiscale potentials, where traditional asymptotic approaches fail. It introduces a localized multiscale finite element method that builds basis functions via a constrained energy-minimization on a coarse mesh, using sparse compression of the Hamiltonian to achieve and a time step independent of ; time evolution is computed through a one-shot eigendecomposition or Jordan-form based solver on the reduced space. The approach yields exponential decay of the basis, enabling localization and a sparse system while preserving gauge invariance and conserving mass and energy. Numerical tests in1D and 2D across various multiscale potentials confirm first-order convergence in and second-order convergence in , demonstrating robustness and efficiency for complex quantum heterostructures and metamaterials. The method offers a practical framework that avoids explicit additive-scale decompositions and suggests potential extensions to stochastic inputs and spintronic device modeling.

Abstract

In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptoics-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis are constructed using sparse compression of the Hamiltonian operator, and thus are "blind" to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is where is the semiclassical parameter and the time stepsize is independent of . Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in and norms, respectively.

Paper Structure

This paper contains 9 sections, 3 theorems, 40 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Multiscale finite element method for the semiclassical Schrödinger equation
  3. Construction of multiscale basis functions
  4. Exponential decay of the multiscale finite element basis functions
  5. Time marching
  6. Eigendecomposition
  7. Jordan canonical form
  8. Numerical examples
  9. Conclusion and discussion

Key Result

Proposition 2.2

Under the resolution condition of the coarse mesh, i.e., eqn:meshcondition, there exist constants $C>0$ and $0<\beta<1$ independent of $H$, such that for any $i=1, 2, ..., N_x$.

Figures (13)

  • Figure 1: Exponentially decaying properties of the multiscale basis function in Example \ref{['example2']} for a sequence of $\varepsilon$ from $1/40$ to $1/160$. Left: $\nabla\phi / ||\phi||_{L_2}$ with respect to the distance to $x=1/2$; Right: $E_{\textrm{relative}}=\dfrac{||\nabla \phi ||_{D}-||\nabla \phi ||_{D_{\ell}}}{\max(||\nabla \phi ||_{D}-||\nabla \phi ||_{D_{\ell}})}$ with respect to the patch size $\ell$. It is observed that the exponentially decaying property of $E_{\textrm{relative}}$ with respect to $\ell$ is independent of $\varepsilon$, which implies the sharpness of \ref{['eqn:exponentialdecay']}.
  • Figure 2: Profiles of numerical and exact density functions for Example \ref{['example1']} when $H/\varepsilon=1/16$ and $\varepsilon=1/40$. Left: position density; Right: energy density.
  • Figure 3: Profiles of numerical and exact density functions for Example \ref{['example1']} when $H/\varepsilon =1/16$ and $\varepsilon=1/256$. Left: position density; Right: energy density.
  • Figure 4: Profiles of numerical and exact density functions for Example \ref{['example2']} when $H / \varepsilon =1/8$ and $\varepsilon=1/40$. Left: position density; Right: energy density.
  • Figure 5: Profiles of numerical and exact density functions for Example \ref{['example2']} when $H/\varepsilon=1/16$ and $\varepsilon=1/256$. Left: position density; Right: energy density.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 2.2: Exponentially decaying property
  • Proposition 2.3: Gauge invariance
  • Proposition 2.4: Conservation of total mass and total energy
  • proof
  • Example 3.1: 1D case with a periodic potential
  • Example 3.2: 1D case with a multiplicative two-scale potential
  • Example 3.3: 1D case with a layered potential
  • Example 3.4: 2D case with an additive two-scale potential
  • Example 3.5: 2D case with a checkboard potential