Homogenization of the Schrödinger-Type Equations: Operator Estimates with Correctors

Abstract

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.