High-frequency homogenization of nonstationary periodic equations

ABSTRACT

In $L_2\left(\mathbb{R}\right)$, we consider an elliptic differential operator $A_{ϵ}$, $ϵ>0$, of the form $A_{ϵ}=-\frac{\mathrm{d}}{\mathrm{d}x}g\left(x/ϵ\right)\frac{\mathrm{d}}{\mathrm{d}x}+{ϵ}^{-2}V\left(x/ϵ\right)$ with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian $A_{ϵ}$ and for the hyperbolic equation with the operator $A_{ϵ}$, analogs of homogenization problems, related to the edges of the spectral bands of the operator $A_{ϵ}$, are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in $L_2\left(\mathbb{R}\right)$-norm for small ε are obtained.

KEYWORDS:

• Schrödinger-type equations
• hyperbolic equations
• spectral bands
• homogenization
• operator error estimates

MATH:

• 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(s).

Additional information

Funding

This work was supported by Young Russian Mathematics award and the Ministry of Science and Higher Education of the Russian Federation, agreement ${\mathcal{N}}^{\circ }$075-15-2022-287.