Spectral approach to homogenization of nonstationary Schrödinger-type equations

Tatyana Suslina
Date and time
Conference - DimaScat : Scattering Theory and Spectral Asymptotics of Differential Operators - in Honour of Dimitri Yafaev

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we consider a selfadjoint strongly elliptic operator $A_\varepsilon$, $\varepsilon >0$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D})$. Here $g({\mathbf x})$ is a periodic bounded and positive definite matrix-valued function, and $b({\mathbf D})$ is a first order differential operator. We study the behavior of the operator $e^{- i t A_\varepsilon}$, $t\in {\mathbb R}$, for small $\varepsilon$. It is proved that, as $\varepsilon \to 0$, $e^{- i t A_\varepsilon}$ converges to $e^{- i t A^0}$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$ (with a suitable $s$) to $L_2({\mathbb R}^d;{\mathbb C}^n)$. Here $A^0 = b({\mathbf D})^* g^0 b({\mathbf D})$ is the effective operator. In a work by Birman and Suslina (2009), the following sharp order error estimate was obtained: $$ | e^{- i t A_\varepsilon} - e^{- i t A^0} |{H^3({\mathbb R}^d) \to L_2({\mathbb R}^d)} \leqslant (C_1 + C_2|t|)\varepsilon.$$ Also, by interpolation, $| e^{- i t A\varepsilon} - e^{- i t A^0} |{H^s \to L_2} = O( \varepsilon^{s/3})$ for $0 \leqslant s \leqslant 3$ . In a recent work, we obtain more subtle results. From one hand, we confirm that the error estimate is sharp: in the general case the estimate
$$| e^{- i t A
\varepsilon} - e^{- i t A^0} |_{H^s \to L_2} = O( \varepsilon)$$ is not true if $s< 3$. The supporting examples are given. From the other hand, we distinguish conditions on the operator under which the result can be improved

$$|e^{-itA_\varepsilon}-e^{-itA^0}|_{H^2({\mathbb R}^d) \to L_2({\mathbb R}^d)} \leqslant (\tilde{C}_1+\tilde{C}_2|t|)\varepsilon,$$ and then also

$$| e^{- i t A_\varepsilon} - e^{- i t A^0} |_{H^s \to L_2} = O( \varepsilon^{s/2})$$

for $0 \leqslant s \leqslant 2$. In particular, this is the case for the scalar elliptic operator $A_\varepsilon = - {\rm div} , g({\mathbf x}/\varepsilon) \nabla$, where the matrix $g({\mathbf x})$ has real entries. The results are applied to study the behavior of the solution $u_\varepsilon({\mathbf x},t)$ of the Cauchy problem for the Schrödinger-type equation $i \partial_t u_\varepsilon({\mathbf x},t) = (A_\varepsilon u_\varepsilon)({\mathbf x},t)$. Applications to the Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given. The method is based on the scaling transformation, the Floquet-Bloch theory and the analytic perturbation theory.

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