High-frequency homogenization of nonstationary periodic equations

TL;DR

This paper develops high-frequency homogenization techniques for nonstationary periodic equations, providing approximations for solutions of Schrödinger and hyperbolic equations with periodic coefficients at spectral band edges.

Contribution

It introduces a novel high-frequency homogenization approach for nonstationary equations with periodic coefficients, focusing on spectral band edges.

Findings

Derived $L_2$-norm approximations for solutions at small $oldsymbol{ ext{epsilon}}$

Extended homogenization methods to nonstationary Schrödinger and hyperbolic equations

Analyzed spectral band edge effects on solution behavior

Abstract

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