High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation
Mark Aleksandrovich Dorodnyi · Russian Journal of Mathematical Physics · 2023
In $$L_2(\mathbb{R}^d)$$ , we consider an elliptic differential operator $$\mathcal{A}_\varepsilon \! = \! - \operatorname{div} g(\mathbf{x}/\varepsilon) abla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)$$ , $$ \varepsilon > 0$$ , with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian $$\mathcal{A}_\varepsilon$$ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator $$\mathcal{A}_1$$ are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in $$L_2(\mathbb{R}^d)$$ -norm for small $$\varepsilon$$ are obtained. DOI 10.1134/S1061920823040064