Homogenization of the Schrödinger-Type Equations: Operator Estimates with Correctors
Tatiana Aleksandrovna Suslina · Functional Analysis and Its Applications · 2022
In $$L_2(\mathbb R^d;\mathbb C^n)$$ we consider a self-adjoint elliptic second-order differential operator $$A_\varepsilon$$ . It is assumed that the coefficients of $$A_\varepsilon$$ are periodic and depend on $$\mathbf x/\varepsilon$$ , where $$\varepsilon>0$$ is a small parameter. We study the behavior of the operator exponential $$e^{-iA_\varepsilon\tau}$$ for small $$\varepsilon$$ and $$\tau\in\mathbb R$$ . The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation $$i\partial_\tau \mathbf{u}_\varepsilon(\mathbf x,\tau) = - (A_\varepsilon{\mathbf u}_\varepsilon)(\mathbf x,\tau)$$ with initial data in a special class. For fixed $$\tau$$ and $$\varepsilon\to 0$$ , the solution $${\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)$$ converges in $$L_2(\mathbb R^d;\mathbb C^n)$$ to the solution of the homogenized problem; the error is of order $$O(\varepsilon)$$ . We obtain approximations for the solution $${\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)$$ in $$L_2(\mathbb R^d;\mathbb C^n)$$ with error $$O(\varepsilon^2)$$ and in $$H^1(\mathbb R^d;\mathbb C^n)$$ with error $$O(\varepsilon)$$ . These approximations involve appropriate correctors. The dependence of errors on $$\tau$$ is traced.