Homogenization of Schrödinger-type equations


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Abstract

We consider a self-adjoint elliptic operator Aε, ε> 0, on L2(Rd; Cn) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space Hs(Rd;Cn) (with suitable s) to L2(Rd;Cn). Here A0 is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i∂τuε(x, τ) = Aεuε(x, τ).

About the authors

T. A. Suslina

St. Petersburg State University

Author for correspondence.
Email: t.suslina@spbu.ru
Russian Federation, St. Petersburg

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