Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a self-adjoint matrix strongly elliptic second-order differential operator ${\mathcal{A}}_{ϵ}$ with periodic coefficients depending on $\mathbf{x}/ϵ$. We find approximations of the exponential ${\mathrm{e}}^{-\mathrm{i}\tau {\mathcal{A}}_{ϵ}}$, $\tau \in \mathbb{R}$, for small ε in the ($H^s\to L_2$)-operator norm with suitable s. The sharpness of the error estimates with respect to τ is discussed. The results are applied to study the behavior of the solution ${\mathbf{u}}_{ϵ}$ of the Cauchy problem for the Schrödinger-type equation $i{\mathrm{\partial }}_{\tau }{\mathbf{u}}_{ϵ}={\mathcal{A}}_{ϵ}{\mathbf{u}}_{ϵ}+\mathbf{F}$.

Keywords:

• Periodic differential operators
• nonstationary Schrödinger-type equations
• homogenization
• effective operator
• operator error estimates

2010 Mathematics Subject Classification:

• 35B27

Acknowledgements

The author is grateful to T. A. Suslina for helpful discussions and advices.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Supported by Young Russian Mathematics award and Ministry of Science and Higher Education of the Russian Federation, agreement N^o 075-15-2019-1619.